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LOWELL 


HYDRAULIC    EXPERIMENTS. 


BEING   A    SELECTION    FROM 


EXPERIMENTS  ON  HYDRAULIC   MOTORS, 


ON    THE 


FLOW    OF   WATER   OVER   WEIRS,   IN   OPEN   CANALS   OF   UNIFORM   RECTANGULAR 
SECTION,   AND   THROUGH    SUBMERGED   ORIFICES   AND    DIVERGING   TUBES. 

I 

MADE     AT     LOWELL,     MASSACHUSETTS. 

BY 

JAMES    B.    FRANCIS, 
t 

CIVIL    ENGINEER,    MEMBER    OP    THE    AMERICAN    SOCIETY    OP    CIVIL    ENGINEERS    AND     ARCHITECTS, 

FELLOW    OF    THE     AMKRIOAN    ACADEMY    OP    ARTS    AND     SCIENCES,    MEMBER 

OF  THE   AMERICAN  PHILOSOPHICAL  SOCIETY,   BTO. 


SECOND    EDITION, 

KEVISED  AND  ENLARGED,  WITH  MANY  NEW  EXPERIMENTS, 

3Lnb  Illustrate!) 
WITH    TWENTY-THREE     COPPEPV-PLATE    ENGRAVINGS. 


NEW    YORK: 
D.    VAN    NOSTRAND,    192    BROADWAY. 

LONDON:  TRUBNER&  CO. 
1868. 


Entered   according  to  Act  of  Congress,   in   the  year  1868,   by 

JAMES    B.    FRANCIS, 
in  the  Clerk'R  Office  of  the  District  Court  of  the  District  of  Massachusetts. 


PREFACE   TO   THE   SECOND   EDITION. 


SINCE  the  first  edition  of  this  work  appeared,  in  1855,  the  manufacturing  corpora- 
tions at  Lowell,  lessees  of  the  water-power  furnished  by  the  Merrimack  River  at  that 
point,  have  surrendered  their  leases  and  taken  others  containing  new  provisions  for  the 
purpose  of  more  fully  protecting  all  parties  in  the  enjoyment  of  their  respective  rights ; 
this  has  rendered  necessary  a  new  and  elaborate  series  of  experiments  for  the  purpose  of 
perfecting  the  method  of  gauging  the  flow  of  water  in  open  channels  by  the  use  of 
loaded  tubes.  Some  experiments  had  been  made  on  this  subject  at  Lowell  before  the 
publication  of  the  first  edition,  the  principal  results  of  which  were  given ;  the  later  ex- 
periments are,  however,  so  much  more  complete,  and  have  been  made  under  circum- 
stances so  much  more  favorable,  that  it  has  been  found  necessary  to  rewrite,  entirely,  the 
chapter  on  that  subject. 

The  general  use  at  Lowell  of  the  Dlffuser,  an  apparatus  for  utilizing  the  power 
usually  lost  in  turbines,  from  the  water  leaving  them  with  a  considerable  velocity,  has 
created  much  interest  in  Venturi's  tube,  the  action  in  which  involves  the  same  principles 
as  the  Diffuser.  Experiments  on  Venturi's  tube  had  been  previously  made  only  when 
discharging  into  the  air ;  it  appeared  highly  probable  that  greater  results  might  be  ob- 
tained if  the  tube  was  submerged,  so  as  to  discharge  under  water.  Experiments  made 
under  these  circumstances,  and  detailed  at  length  in  this  edition,  indicate  a  considerably 
greater  flow  than  had  been  previously  obtained. 

The  author  takes  this  opportunity  of  acknowledging  his  obligations  to  Mr.  Uriah  A. 
Boyden  of  Boston,  for  useful  suggestions  during  the  last  twenty-five  years,  on  almost 
every  subject  discussed  in  this  volume.  Also  to  Mr.  John  Newell,  now  of  Detroit, 
Michigan,  to  whom  he  is  much  indebted  for  assistance  in  the  execution  and  reduction 
of  some  of  the  most  important  series  of  experiments,  and  to  whose  fidelity  the  precision 
attained  in  the  results  is  in  no  small  degree  due.  Also  to  Mr.  Joseph  P.  Frizell,  now 
of  Davenport,  Iowa,  to  whom  he  is  indebted  for  assistance  in  some  points  involving  the 
higher  mathematics. 

» 

LOWKLI,,   MASS.,  March,   18G8. 


TABLE    OF    CONTENTS. 


INTRODUCTION. 

PART    I. 

EXPERIMENTS  ON  HYDRAULIC  MOTORS. 

Number  of  „ 

the  Article. 

EXPERIMENTS  UPON  THE  TKEMONT  TURBINE, 1 

1-17.         Introduction, 1 

18—35.  Description  of  the  Turbine,          ............7 

36—47.         Description  of  the  Apparatus  used  in  the  Experiments, 14 

48—53.  Mode  of  conducting  the  Experiments,           .         .         .         .         .         .         .         .         .         .19 

54-74.  Description  of  Table  II.,  containing  the  Experiments  upon  the  Turbine  at  the  Tremont  Mills,     25 

75-82.  Description  of  the  Diagram  representing  the  Experiments,       .         .         .         .         .         .         36 

83-88.  Path  described  by  a  Particle  of  Water  in  passing  through  the  Wheel,       .         .         .         .39 

89-98.    RULES  FOR  PROPORTIONING  TURBINES, 44 

99-109.     EXPERIMENTS    ON    A     MODEL    OF    A    CENTRE-VENT    WATER-WHEEL,    WITH    STRAIGHT 

BUCKETS, 55 

110-119.     EXPERIMENTS    ON  THE   POWER   OF   A    CENTRE- VENT   WATER-WHEEL,   AT    THE    BOOTT 

COTTON-MILLS, 61 


PART    II. 

EXPERIMENTS   ON   THE   FLOW   OF   WATER  OVER  WEIRS,  AND   IN   SHORT 

RECTANGULAR    CANALS. 
i 

EXPERIMENTS  ON  THE  FLOW  OF  WATKR  OVER  WEIRS, 71 

120-125.         Introduction, .71 

126-135.         Experiments  made  at  the  Tremont  Turbine,  on  the  Flow  of  Water  over  Weirs,    .  76 


VI 


CONTENTS. 


136.  Experiments  on    the  Flow  of    Water   over  Weirs,  made   at    the  Centre-Vent  Wheel  for 

moving  the  Guard  Gates  of  the  Northern  Canal,          .......       96 

137.  Experiments  on  the  Effect  produced   on  the  Flow  of  Water  over  Weirs,  by   the  Height 

of  the  Water  on  the  Downstream  Side,         .........       99 

138-147.  Experiments  on  the  Flow  of  Water  over  Weirs,  made  at  the  Lower  Locks,  Lowell,  .  103 
148-159.  Description  of  Table  XIII.,  containing  the  Details  of  the  Experiments  on  the  Flow  of 

Water  over  Weirs  made  at  the  Lower  Locks,  Lowell,  in  October  and  November,  1852,  112 
160—163.  Comparison  of  the  proposed  Formula  with  the  Results  obtained  by  previous  Experimenters,  126 
164.  Precautions  to  be  observed  in  the  application  of  the  proposed  Formula,  .  .  .  133 

165—166.  Experiments  on  the  Discharge  of  Water  over  a  Dam,  of  the  same  Section  as  that  erected 

by  the  Essex  Company,  across  the  Merrimack  River  at  Lawrence,  Massachusetts,  .  136 
167-175.  Experiments  to  ascertain  the  Effect  of  taking  the  Depths  upon  a  Weir,  by  means  of 

Pipes  opening  near  the  Bottom  of  the  Canal,        ........     137 

176.  Formula  for  the  Discharge  over  Weirs  in  which  the  crest  is  not  horizontal.  Formula 

for  the  Discharge  over  Weirs  for  any  Latitude  or  Height  above  the  Sea   .          .          143 

A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER  IN  OPEN  CANALS  OF  UNIFORM 

RECTANGULAR  SECTION,  AND  OF  SHORT  LENGTH. 

177-179.         Arrangements  at  Lowell  for  the  Distribution  of  the  Water-Power  among  the  several  Lessees     146 

180.  Method  of  Gauging  the  Water  drawn  at  one  of  the  Cotton  Mills  of  the  Hamilton  Manu- 

facturing Company  in  1830  ...........     148 

181.  Experiments  of  Messrs.  Baldwin,  Whistler,  and  Storrow  in   1841   and   1842       .          .          148 
182-198.         Method  of  Gauging  the  Flow  of  Water  in  Open  Canals  by  means  of  Loaded  Poles  or 

Tubes 155 

199-225.  Experiments  made  to  determine  a  Formula  of  Correction  for  Gaugings  in  Open  Canals, 

by  means  of  Loaded  Poles  or  Tubes            .          .          .          .          .          .          .          .  169 

226-238.  Formula  of  Correction  for  Gaugings  made  with  Loaded  Poles  or  Tubes  .  .  .  191 
239-246.  Application  of  the  Method  of  Gauging  Streams  of  Water  by  means  of  Loaded  Poles  or 

Tubes  201 


EXPERIMENTS  ON  THE  FLOW  OF  WATER  THROUGH  SUBMERGED  ORIFICES  AND 

DIVERGING  TUBES. 

247-250.         Former  Experiments  on  this  Subject 209 

251-254.         Description  of  the  Apparatus  used  in  the  New  Experiments     .          .          .          .          .  212 

255-257.         Mode  of  conducting  the  Experiments  .........  216 

258-260.         Description  of  the  Experiments 222 

261-269.         Deductions  from  the  Experiments       .          .          . 223 

270.         Description  of  a  Turbine  Water- Wheel  of  700  Horse  Power 230 


\  - 


CONTENTS.  vii 

TABLES. 

PART    I. 

Number  of  p 

the  Table. 

I.     Weight  of  a  Cubic  Foot  of  pure  Water,  at  different  Temperatures,       .....         29 
II.     Experiments  upon  the  Turbine  at  the  Tremont  Mills,  in  Lowell,  Massachusetts,       .         .         .32 

III.  Successive  Steps  in   the   Calculation  for  the   Path  of  the    Water  in  Experiment  30   on   the 

Tremont  Turbine,         ..............         41 

IV.  Table  for  Turbines  of  different  Diameters,  operating  on  different  Falls,     .         .         .         .         .53 
V.     Experiments  on  a  Model  of  a  Centre- Vent  Water- Wheel, 58 

VI.     Experiments  on  the  Boott  Centre-Vent  Water- Wheel,      \.  •  \ 66 

VII.     Successive    Steps   in  the   Calculation   for  the  Path  of  the  Water  in    Experiment   30,  on   the 

Boott  Centre-Vent  Water- Wheel, 70 


PART     II. 

VIII.     Experiments  made  at  the  Tremont  Turbine,  for  the  purpose  of  testing  the  Method  of  reduc- 
ing the  Depths  on  the  Weir  to  a  uniform  Fall, 83 

IX.     Experiments  made  at  the  Tremont  Turbine,  which  were  repeated  under  identical  circumstances,      84 
X.     Experiments  on  the  Flow  of  Water  over  Weirs,  made  at  the  Tremont  Turbine,  .         .         88 

XI.     Experiments  on  the  Flow  of  Water  over  Weirs,  made  at  the  Centre- Vent  Wheel  for  mov- 
ing the  Guard  Gates  of  the  Northern  Canal,  at  Lowell,  Massachusetts,       .         .         .         .98 
XII.     Experiments  on  the  Effect  produced  on  the  Flow  of  Water  over  Weirs,  by  the  Height  of 

the  Water  on  the  Downstream  Side, 102 

XIII.  Experiments  on  the  Flow  of  Water  over  Weirs,  made  at  the  Lower  Locks,  Lowell,  in  Octo- 

ber and  November,  1852, 122 

XIV.  Comparison  of  the  proposed  Formula  with  the  Experiments  of  Poncelet  and  Lesbros,          .         128 
XV.     Comparison  of  the  proposed  Formula  with  the  Experiments  of  Castel, 130 

XVI.  Experiments  on  the  Discharge  of  Water  over  a  Dam  of  the  same  Section  as  that  erected  by 

the  Essex  Company,  across  the  Merrimack  River  at  Lawrence,  Massachusetts,        .         .         137 

XVII.  Experiments    made    at  the  Lower  Locks,  to  determine   the  Corrections  to  be  applied  to  the 

Readings  of  the  Hook  Gauges,      . 140 

XVIH.     Experiments  to  ascertain  the  Effect  of  taking  the  Depths  upon  a  Weir,  by  means  of  Pipes 

opening  near  the  Bottom  of  the  Canal,     ..........  142 

A.  B.  C.     Experiments  of  Messrs.  Baldwin,  Whistler,  and  Storrow,  made  for  the  Purpose  of  finding 

the  Ratio  between  the  Mean  and  Surface  Velocities  in  certain  Open  Canals  .          .          .152 

XIX.    Data  and  Computed  Results  of  four  Experiments  with  Loaded  Tubes  .         .         ...  168 

XX.     Observations  in  Experiment  No.  1,  Table  XXII 176 


Vlll 


CONTENTS. 


XXI.  Comparison  of  the   Height  of  the  Tops  of  the  Weirs  with   the   Point  of  the   Stationary 

Hook 179 

XXII.  Experiments  from  which  the  Formula  of  Correction  for  Flume  Measurements  is  determined  186 

XXIII.  Mean  Results  of  the  Experiments  in  Table  XXII.  arranged  according  to  Velocities     .          .192 

XXIV.  Mean  Results  of  the  Experiments  in  Table  XXII.  arranged  according  to  Lengths  of  Tubes  195 

XXV.  Miscellaneous  Experiments  at  the  Tremont  Measuring  Flume       .....  198 

XXVI.  Gauge  of  the  Quantity  of  Water  passing  the  Boott  Measuring  Flume,  May  17,  1860     .     .  204 

XXVII.  Experiments  on  the  Flow  of  Water  through  Submerged  Tubes  and  Orifices           .          .  218 

XXVIII.    Velocities  of  Floats  in  Measuring  Flumes 233 

XXIX.  Corrections  for  Flume  Measurements         .........  241 

XXX.    Velocities  due  Heads  for  every  0.01  Foot  up  to  49.99  Feet 242 


INTRODUCTION. 


THE  northern  regions  of  the  United  States  of  North  America,  probably  possess 
a  greater  amount  of  water-power  than  any  other  part  of  the  world  of  equal 
extent,  and  the  active  and  inventive  genius  of  the  American  people,  combined 
with  the  very  high  price  of  labor,  has  had  a  powerful  influence  in  bringing  this 
power  into  use.  Nevertheless,  the  water-power  is  so  vast,  compared  with  the  pop- 
ulation, that  only  a  small  portion  of  it  has,  up  to  this  time,  been  applied  to  the 
purposes  of  man.  It  was  estimated,  not  long  since,  that  the  total  useful  effect 
derived  from  water-power  in  France,  was  about  20,000  horse-power.  An  amount 
of  power  far  exceeding  this,  is  already  derived  from  the  Merrimack  Eiver  and 
its  branches,  in  Massachusetts  and  New  Hampshire.  What  must  be  the  amount 
of  the  population  and  wealth  of  the  Northern  States,  when  the  other  rivers  that 
water  them  are  equally  improved? 

One  of  the  earliest  and  most  successful  efforts  to  bring  into  use,  in  a  sys- 
tematic manner,  one  of  the  larger  water-powers,  was  made  at  Lowell  in  Massa- 
chusetts ;  where,  in  1821,  a  number  of  farms  situated  near  Pawtucket  Falls  on 
the  Merrimack  River,  were  purchased  by  several  capitalists  of  Boston,  who  obtained 
a  charter  from  the  State  of  Massachusetts  under  the  name  of  The  Merrimack 
Manufacturing  Company.  In  1826,  the  property  was  transferred  to  the  Proprietors 
of  ike  Locks  and  Canals  on  Merrimack  River,  a  corporation  chartered  in  1792  for 
the  purpose  of  improving  the  navigation  of  the  Merrimack  River.  Previously  to 
the  transfer,  the  Merrimack  Manufacturing  Company  had  erected  a  dam  of  about 
950  feet  in  length,  at  the  head  of  Pawtucket  Falls,  and  had  also  enlarged  the 
Pawtucket  Canal,  which  was  originally  constructed,  previously  to  the  year  1800, 
by  the  Proprietors  of  the  Locks  and  Canals  on  Merrimack  River,  for  the  purposes 


x  INTRODUCTION. 

of  navigation.  Subsequently  to  the  enlargement,  however,  this  canal  has  been 
used  both  for  purposes  of  navigation,  and  to  supply  water  to  the  wheels  of 
numerous  manufacturing  establishments. 

The  dam  at  the  head  of  Pawtucket  Falls,  in  the  ordinary  state  of  the  river, 
deadens  the  current  of  the  river  for  about  18  milee),  forming,  in  low  water,  a 
reservoir  of  about  1120  acres;  this  extensive  reservoir  is  of  great  value  in  very 
low  stages  of  the  river,  as  it  affords  space  for  the  accumulation  of  the  flow  of 
the  river  during  the  night,  when  the  manufactories  are  not  in  operation.  This 
accumulation  is  subsequently  drawn  off,  together  with  the  natural  flow  of  the 
river,  during  the  usual  working  hours. 

The  total  fall  of  the  Merrimack  River  at  Pawtucket  Falls,  in  ordinary  low 
water,  is  about  35  feet,  of  which  about  2  feet  is  lost  in  consequence  of  the 
descent  in  the  canals,  leaving  a  net  fall  of  about  33  feet.  About  £  of  the  water 
is  used  on  the  entire  fall,  and  the  remainder  is  used  twice  over,  on  falls  of  about 
14  and  19  feet  respectively.  The  water-power  has  been  granted  by  the  Propri- 
etors of  the  Locks  and  Canals  on  Merrimack  Elver,  in  definite  quantities  called 
Mill  Pmvers,  which  are  equivalent  to  a  gross  power  of  a  little  less  than  100 
horse-power  each.  Grants  have  been  made  to  eleven  manufacturing  companies, 
who  have  an  aggregate  capital,  somewhat  exceeding  thirteen  millions  of  dollars. ' 
Thus,  to  the  Merrimack  Manufacturing  Company,  there  have  been  granted  24-f 
mill  powers,  each  of  which  consists  of  the  right  to  draw,  for  15  hours  per  day, 
25  cubic  feet  of  water  per  second  on  the  entire  fall.  Up  to  this  time,  there 
have  been  granted  at  Lowell  139-ij-J-  mill  powers,  or  a  total  quantity  of  water 
equal  to  3595.933  cubic  feet  per  second.  A  large  portion  of  this  water  is  used 
on  turbines  of  a  very  superior  description,  and  nearly  all  the  remainder,  on  breast 
wheels  of  good  construction,  a.  portion  of  which,  however,  do  not  use  quite  the 
whole  of  the  fall  on  which  they  are  placed.  We  may,  however,  assume  that, 
upon  an  average,  a  useful  effect  is  derived,  equal  to  f  of  the  total  power  of 
the  water  expended.  Calling  the  fall  33  feet,  and  the  weight  of  a  cubic  foot 
of  water  62.33  pounds,  we  shall  have  for  the  effective  power  derived  from  the 
water-power  granted  by  the  Proprietors  of  the  Locks  and  Canals  on  Merrimack 
River  at  Lowell, 

3595.933  X  62.33  X  33  X  3 

550"  ==  8965.4  horse-power. 


INTRODUCTION.  xi 

s 

In  consequence  of  the  success  attending  the  improvement  of  the  water-power 
at  Lowell,  several  other  extensive  water-powers  in  New  England  have  been  brought 
into  use  in  a  similar  manner.  Some  of  these  undertakings  have  been  quite  suc- 
cessful, whilst  with  others,  as  yet  only  partially  developed,  the  success  has  not 
been  so  decided. 

The  great  abundance  of  water-power  in  this  country  has  had  a  strong  ten- 
dency to  encourage  its  extravagant  use ;  the  machines  used  in  the  manufactories 
are  usually  great  consumers  of  power ;  the  ability  of  a  machine  to  turn  off  the 
greatest  quantity  of  work  with  the  least  manual  labor,  and  in  the  least  time,  has 
been  the  point  mainly  considered;  and  whether  it  required  a  greater  or  less 
amount  of  power,  has  been  a  secondary  consideration. 

The  engineering  operations  connected  with  the  water-power  at  Lowell,  have 
frequently  demanded  more  definite  information  on  certain  points  in  hydraulics, 
than  was  to  be  found  in  any  of  the  publications  relating  to  that  science ;  and 
hence  has  arisen  the  necessity,  from  time  to  time,  of  making  special  experiments 
to  supply  the  required  information.  Whenever  such  emergencies  have  arisen,  the 
officers  who  have  the  general  care  of  the  interests  of  the  several  corporations, 
with  a  liberality  founded  on  enlarged  views  of  the  true  interests  of  the  bodies 
they  represent,  have  always  been  willing  to  defray  such  expenses  as  were  neces- 
sary, in  order  that  the  experiments  might  be  made  in  a  satisfactory  manner. 

The  experiments  recorded  in  the  following  pages,  are  a  selection  from  those 
made  by  the  author,  in  the  discharge  of  his  duty,  as  the  Engineer  of  the  Cor- 
porations at  Lowell.  They  may  be  divided  into  two  classes,  namely,  first,  those 
on  hydraulic  motors,  and,  second,  those  on  the  flow  of  water  over  weirs,  and  in 
short  rectangular  canals.  Combined  with  the  description  of  the  experiments,  there 
are  also  given  some  other  investigations,  which  may  appear  somewhat  out  of 
place,  but  which,  from  their  utility  or  novelty,  will  be  found  interesting  to  many 
persons  who  have  cultivated  the  science  of  hydraulics. 

The  unit  of  length  adopted  in  this  work,  is  the  English  foot  according  to  a 
brass  standard  measure  made  by  Gary  of  London,  now  in  the  possession  of  the 
Lowell  Machine  Shop. 


HYDRAULIC     EXPERIMENTS, 


PART    I. 

EXPERIMENTS    ON    HYDRAULIC    MOTORS. 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 

1.  UNTIL  within  a  few  years,  the  water-wheels  in  use  in  the  principal  manufac- 
turing establishments  in  New  England,  were  what  are  there  generally  called  breast 
wheels,  sometimes  known  also  by  the  name  of  pitch  back  ivheels.  They  are  the  same  in 
principle  as  the  overshot-ivheel,  the  useful  effect  being  produced,  almost  entirely,  by  the 
simple  weight  of  the  water  in  the  buckets,  and  differing  only  from  the  ovcrsJiot-ivheel 
in  this,  that  the  water  is  not  carried  entirely  over  the  top  of  the  wheel,  but  is  let 
into  the  buckets  near  the  top,  but  on  the  opposite  side  from  that  adopted  for  the 
overshot-wheel.  An  apron,  fitting  as  closely  as  practicable  to  the  wheel,  is  used  to 
prevent  the  water  leaving  the  buckets,  until  it  reaches  very  nearly  the  bottom  of 
the  wheel. 

In  Lowell,  these  wheels  have  been  constructed  principally  of  wood,  many  of 
them  of  very  large  dimensions.  Those  in  the  mills  of  the  Merrimack  Manufacturing 
Company,  for  instance,  are  thirty  feet  in  diameter,  with  buckets  twelve  feet  long.  Four 
of  the  mills  belonging  to  this  company,  have  two  such  wheels  in  each  of  them. 

Until  the  year  1844,  the  breast  wheel,  as  above  described,  was  considered  here 
the  most  perfect  wheel  that  could  be  used.  Much  prejudice  existed  here,  as  elsewhere, 
against  the  reaction  wheels  ;  a  great  number  of  which  had,  however,  been  used 
throughout  the  country,  in  the  smaller  mills,  and  with  great  advantage ;  for,  although 
they  usually  gave  a  very  small  effect  in  proportion  to  the  quantity  of  water  expended, 
their  cheapness,  the  small  space  required  for  them,  their  greater  velocity,  being  less 

1 


2  EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 

impeded  by  backwater,  and  not  requiring  expensive  wheelpits  of  masonry,  were  very 
important  considerations ;  and  in  a  country  where  water  power  is  so  much  more 
abundant  than  capital,  the  economy  of  money  was  generally  of  greater  importance 
than  the  saving  of  water. 

A  vast  amount  of  ingenuity  has  been  expended  by  intelligent  millwrights,  on 
these  wheels ;  and  it  was  said,  several  years  since,  that  not  less  than  three  hundred 
patents  relating  to  them,  had  been  granted  by  the  United  States  Government.  They 
continue,  perhaps  as  much  as  ever,  to  be  the  subject  of  almost  innumerable  modifica- 
tions. Within  a  few  years,  there  has  been  a  manifest  improvement  in  them,  and  there 
are  now  several  varieties  in  use,  in  which  the  wheels  themselves  are  of  simple  forms, 
and  of  single  pieces  of  cast-iron,  giving  a  useful  effect  approaching  sixty  per  cent,  of 
the  power  expended. 

2.  The  attention  of  American  engineers  was  directed  to  the  improved  reaction 
water-wheels  in  use  in  France  and  other  countries  in  Europe,  by  several  articles  in  the 
Journal   of  the  Franklin  Institute ;    and   in   the   year   1843,  there   appeared   in   that 
journal,  from  the  pen  of  Mr.  Ellwood  Morris,  an  eminent  engineer  of  Pennsylvania, 
a  translation  of  a  French  work,  entitled,  Experiments  on  water-ivlieels  having  a  vertical  axis, 
catted  turbines,  by  Arthur  Morin,   Captain  of  Artillery,  etc.  etc.      In   the   same  journal,  Mr. 
Morris  also  published  an  account  of  a  series  of  experiments,  by  himself,  on  two  turbines 
constructed  from  his  own  designs,  and  then  operating  in  the  neighborhood  of  Phila- 
delphia. 

The  experiments  on  one  of  these  wheels,  indicate  a  useful  effect  of  seventy-five  per 
cent,  of  the  power  expended,  a  result  as  good  as  that  claimed  for  the  practical  effect 
of  the  best  overshot-wheels,  which  had,  heretofore,  in  this  country,  been  considered 
unapproachable,  in  their  economical  use  of  water. 

3.  In   the   year    1844,   Uriah  A.  Boyden,  Esq.,  an   eminent   hydraulic   engineer 
of  Massachusetts,  designed  a  turbine  of  about  seventy-five  horse-power,  for  the  Pick- 
ing House   of  the  Appleton   Company's   cotton-mills,  at   Lowell,  in  Massachusetts,  in 
which  wheel,  Mr.  Boyden  introduced  several  improvements,  of  great  value. 

The  performance  of  the  Appleton  Company's  turbine,  was  carefully  ascertained 
by  Mr.  Boyden,  and  its  effective  power,  exclusive  of  that  required  to  carry  the  wheel 
itself,  a  pair  of  bevel  gears,  and  the  horizontal  shaft  carrying  the  friction  pulley  of  a 
Prony  dynamometer,  was  found  to  be  seventy-eight  per  cent,  of  the  power  expended. 

4.  In  the  year  1846,  Mr.   Boyden  superintended  the  construction  of  three  tur- 
bines of  about  one  hundred  and  ninety  horse-power  each,  for  the  same  company.     By 
the  terms  of  the    contract,  Mr.  Boyden's   compensation   depended   upon  the  perform- 
ance of  the  turbines,  and  it  was  stipulated  that  two  of  them  should  be  tested.     The 
contract  also  contained  the  following  clause,  "and  if  the  mean   power   derived  from 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE.  3 

these  turbines  be  seventy-eight  per  cent,  of  the  power  of  water  expended,  the  Apple- 
ton  Company  to  pay  me  twelve  hundred  dollars  for  my  services,  and  patent  rights 
for  the  apparatus  for  these  mills;  and  if  the  power  derived  be  greater  than  seventy- 
eight  per  cent.,  the  Appleton  Company  to  pay  me,  in  addition  to  the  twelve  hundred 
dollars,  at  the  rate  of  four  hundred  dollars  for  every  one  per  cent,  of  power,  obtained 
above  seventy-eight  per  cent."  In  accordance  with  the  contract,  two  of  the  turbines 
were  tested,  a  very  perfect  apparatus  being  designed  by  Mr.  Boyden  for  the  purpose, 
consisting,  essentially,  of  a  Prony  dynamometer  to  measure  the  useful  effects,  and 
a  weir  to  gauge  the  quantity  of  water  expended. 

5.  A  great  improvement  in  the  mode  of  conducting  hydraulic  experiments  was 
here  adopted,  in  making  each  set  of  observations  continuous,  the  time  of  each  obser- 
vation being  noted ;  thus,  the  observer  who  noted  the  height  of  the  water  above  the 
wheel,  recorded  regularly,  say  every  thirty  seconds,  the  time  and  the  height ;  and  so 
•with  the  other  observers,  the  recorded  times  furnishing  the  means  of  afterwards  identi- 
fying simultaneous  observations. 

6.  The  observations   were  put   into  the   hands  of  the    author,  for   computation, 
who  found  that  the  mean  maximum  effective  power  of  the  two  turbines  tested,  was 
eighty-eight  per  cent,  of  the  power  of  the  water  expended. 

According  to  the  terms  of  the  contract,  this  made  the  compensation  for  engineering 
services,  and  patent  rights  for  these  three  wheels,  amount  to  fifty-two  hundred  dollars, 
which  sum  was  paid  by  the  Appleton  Company  without  objection. 

7.  These  turbines  have  now  been  in  operation  about  eight  years,  and  their  per- 
formance has  been,  in  every  respect,  entirely  satisfactory.      The  iron-work  for  these 
wheels  was  constructed  by  Messrs.  Gay  and  Silver,  at  their  machine  shop  at  North 
Chelmsford,  near  Lowell ;  the  workmanship  was  of  the  finest  description,  and  of  a  deli- 
cacy and  accuracy  altogether  unprecedented  in  constructions  of  this  class. 

8.  These  wheels,  of  course,  contained  Mr.  Boyden's  latest  improvements,  and   it 
was  evidently  for  his  pecuniary  interest  that  the  wheels  should  be  as  perfect  as  possible, 
without  much  regard  to  cost.     The  principal  points  in  which  one  of  them  differs  from 
the  constructions  of  Fourneyron,  are  as  follows. 

9.  The  wooden  flume,  conducting  the  water  immediately  to  the  turbine,  is  in  the  form  of 
an  inverted  truncated  cone,  the  ivater  being  introduced  into  the  upper  part  of  the  cone,  on  one 
side  of  the  axis  of  the  cone   (which  coincides  with  the  axis  of  the  turbine)  in  such  a  manner, 
that  the  water,  as   it  descends   in  the   cone,  Jias  a  gradually  increasing  velocity,  and  a  spiral 
motion ;  the  horizontal  component  of  the  spiral  motion  being  in  the  direction  of  the  motion  of  the 
^vheel.     This  horizontal  motion  is  derived   from  the  necessary  velocity  with  which  the 
water  enters  the  truncated  cone ;  and  the  arrangement  is  such  that,  if  perfectly  propor- 
tioned, there  would  be  no  loss  of  power  between  the  nearly  still  water  in  the  principal 


4  EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 

penstock  and  the  guides  or  leading  curves  near  the  wheel,  except  from  the  friction 
of  the  water  against  the  walls  of  the  passages.  It  is  not  to  be  supposed  that  the 
construction  is  so  perfect  as  to  avoid  all  loss,  except  from  friction ;  but  there  is,  without 
doubt,  a  distinct  advantage  in  this  arrangement  over  that  which  had  been  usually 
adopted,  and  where  no  attempt  had  been  made  to  avoid  sudden  changes  of  direction 
and  velocity. 

10.  The  guides,  or  leading  curves,  are  not  perpendicular,  lid  a  little  inclined  lachvards 
from  the  direction  of  the  motion  of  the  ivheel,  so  that  the  water,  descending  with  a  spiral  motion, 
meets  only  the  edges  of  the  guides.     This  leaning  of  the  guides  has  also  another  valuable 
effect;  when  the  regulating  gate  is  raised  only. a  small  part  of  the  height  of  the  wheel, 
the  guides  do  not  completely  fulfil  their  office  of  directing  the  water,  the  water  entering 
the  wheel  more  nearly  in  the  direction  of  the  radius,  than  when  the  gate  is  fully  raised ; 
by  leaning  the  guides,  it  will  be  seen  that  the  ends  of  the  guides,  near  the  wheel,  are 
inclined,  the  bottom  part  standing  further  forward,  and  operating  more  efficiently  in 
directing  the  water,  when  the  gate  is  partially  raised,  than  if  the  guides  were  perpen- 
dicular. 

11.  In  Fourneyron's  constructions,  a  garniture  is  attached  to  the  regulating  gate, 
and  moves  with   it,  for  the  purpose  of  diminishing  the  contraction ;   this,  considered 
apart  from  the  mechanical  difficulties,  is  probably  the  best  arrangement ;  to  be  perfect, 
however,  theoretically,  this  garniture  should  be  of  different  forms  for  different  heights 
of  gate ;  but  this  is  evidently  impracticable. 

In  the  Appleton  Turbine,  the  garniture  is  attached  to  the  guides,  the  gate  (at  least  the  lower 
part  of  it]  being  a  simple  thin  cylinder.  By  this  arrangement,  the  gate  meets  with  much 
less  obstruction  to  its  motion  than  in  the  old  arrangement,  unless  the  parts  are  so 
loosely  fitted  as  to  be  objectionable ;  and  it  is  believed  that  the  coefficient  of  effect, 
for  a  partial  gate,  is  proportionally  as  good  as  under  the  old  arrangement. 

12.  On  the  outside  of  the  wheel  is  fitted  an  apparatus  named,  ly  Mr.  Boydcn,  the  Diffuser. 
The  object  of  this  extremely  interesting  invention,  is  to  render  useful  a  part  of  the 
power  otherwise  entirely  lost,  in  consequence   of  the  water  leaving  the  wheel   with 
a  considerable  velocity.     It  consists,  essentially,  of  two  stationary  rings  or  discs,  placed 
concentrically  with  the  wheel,  having  an  interior  diameter  a  very  little  larger  than 
the  exterior  diameter  of  the  wheel ;   and  an  exterior  diameter  equal  to  about  twice 
that  of  the  wheel ;  the  height  between  the  discs,  at  their  interior  circumference,  is 
a  very  little  greater  than  that  of  the  orifices  in  the  exterior  circumference  of  the  wheel, 
and  at  the  exterior  circumference  of  the  discs,  the  height  between  them  is  about  twice 
as  great  as  at  the  interior  circumference ;    the  form  of  the  surfaces  connecting  the 
interior  and  exterior  circumferences  of  the  discs,  is  gently  rounded,  the  first  elements 
of  the  curves,  near  the  interior  circumferences,  being  nearly  horizontal.     There  is  con- 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE.  5 

sequently,  included  between  the  two  surfaces,  an  aperture  gradually  enlarging  from 
the  exterior  circumference  of  the  wheel,  to  the  exterior  circumference  of  the  diffuser. 
When  the  regulating  gate  is  raised  to  its  full  height,  the  section,  through  which  the 
water  passes,  will  be  increased  by  insensible  degrees,  in  the  proportion  of  one  to  four, 
and  if  the  velocity  is  uniform  in  all  parts  of  the  diffuser  at  the  same  distance  from  the 
wheel,  the  velocity  of  the  water  will  be  diminished  in  the  same  proportion ;  or  its 
velocity  on  leaving  the  diffuser,  will  be  one  fourth  of  that  at  its  entrance.  By  the 
doctrine  of  living  forces,  the  power  of  the  water  in  passing  through  the  diffuser  must, 
therefore,  be  diminished  to  one  sixteenth  of  the  power  at  its  entrance.  It  is  essential 
to  the  proper  action  of  the  diffuser,  that  it  should  be  entirely  under  water ;  and  the 
power  rendered  useful  by  it,  is  expended  in  diminishing  the  pressure  against  the  water 
issuing  from  the  exterior  orifices  of  the  wheel;  and  the  effect  produced,  is  the  same 
as  if  the  available  fall  under  which  the  turbine  is  acting,  is  increased  a  certain  amount. 
It  appears  probable  that  a  diffuser  of  different  proportions  from  those  above  indicated, 
would  operate  with  some  advantage  without  being  submerged.  It  is  nearly  always 
inconvenient  to  place  the  wheel  entirely  below  low-water-mark ;  up  to  this  time, 
however,  all  that  have  been  fitted  up  with  a  diffuser,  have  been  so  placed ;  and,  indeed, 
to  obtain  the  full  effect  of  a  fall  of  water,  it  appears  essential,  even  when  a  diffuser 
is  not  used,  that  the  wheel  should  be  placed  below  the  lowest  level  to  which  the 
water  falls  in  the  wheelpit,  when  the  wheel  is  in  operation. 

The  action  of  the  diffuser  depends  upon  similar  principles  to  that  of  diverging 
conical  tubes,  which,  when  of  certain  proportions,  it  is  well  known,  increase  the 
discharge ;  the  author  has  not  met  with  any  experiments  on  tubes  of  this  form, 
discharging  under  water,  although,  there  is  good  reason  to  believe,  that  tubes  of  greater, 
length  and  divergency  would  operate  more  effectively  under  water,  than  when  discharg- 
ing freely  in  the  air ;  and  that  results  might  be  obtained,  that  are  now  deemed  impossible 
by  most  engineers. 

Experiments  on  the  same  turbine,  with  and  without  a  diffuser,  show  a  gain  in 
the  coefficient  of  effect,  due  to  the  latter,  of  about  three  per  cent.  By  the  principles 
of  living  forces,  and  assuming  that  the  motion  of  the  water  is  free  from  irregularity, 
the  gain  should  be  about  five  per  cent.  The  difference  is  due,  in  part  at  least,  to  the 
unstable  equilibrium  of  water,  flowing  through  expanding  apertures ;  this  must  interfere 
with  the  uniformity  of  the  velocities  of  the  fluid  streams,  at  equal  distances  from 
the  wheel. 

13.  Suspending  the  wheel  from  the  top  of  the  vertical  shaft,  instead  of  running  it  on  a  step 
at  the  bottom.  This  had  been  previously  attempted,  but  not  with  such  success  as  to 
warrant  its  general  adoption.  It  has  been  accomplished  with  complete  success  by 
Mr.  Boyden,  whose  mode  is,  to  cut  the  upper  part  of  the  shaft  into  a  series  of  necks, 


G  EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 

and  to  rest  the  projecting  parts  upon  corresponding  parts  of  a  box.  A  proper  fit  is 
secured  by  lining  the  box,  which  is  of  cast-iron,  with  babbitt  metal,  a  soft  metallic 
composition  consisting,  principally,  of  tin ;  the  cas1>iron  box  is  made  with  suitable  pro- 
jections and  recesses  to  support  and  retain  the  soft  metal,  which  is  melted  and  poured 
into  it,  the  shaft  being  at  the  same  time  in  its  proper  position  in  the  box.  It  will 
readily  be  seen  that  a  great  amount  of  bearing  surface  can  be  easily  obtained  by  this 
mode,  and  also,  what  is  of  equal  importance,  it  may  be  near  the  axis ;  the  lining 
metal,  being  soft,  yields  a  little  if  any  part  of  the  bearing  should  receive  a  great 
excess  of  weight.  The  cast-iron  box  is  suspended  on  gimbals,  similar  to  those  usually 
adopted  for  mariners'  compasses  and  chronometers,  which  arrangement  permits  the 
box  to  oscillate  freely  in  all  directions,  horizontally,  and  prevents,  in  a  great  measure, 
all  danger  of  breaking  the  shaft  at  the  necks,  in  consequence  of  imperfections  in  the 
workmanship,  or  in  the  adjustments.  Several  years'  experience  has  shown,  that  this 
arrangement,  carefully  constructed,  is  all  that  can  be  desired ;  and  that  a  bearing  thus 
constructed,  is  as  durable,  and  can  be  as  readily  oiled,  and  taken  care  of,  as  any  of  the 
ordinary  bearings  in  a  manufactory. 

14.  The  buckets  are  secured  to  the  crowns  of  the  wheel  in  a  novel,  and  much 
more  perfect  manner,  than  had  been  previously  used ;   the  crowns  are  first  turned  to 
the  required  form,  and  made  smooth ;  by  ingenious  machinery  devised  for  the  purpose, 
grooves  are  cut  with  great  accuracy  in   the    crowns,  of  the  exact   curvature  of  the 
buckets ;  mortices  are  cut  through  the  crowns,  in  several  places  in  each  groove ;  the 
buckets,  or  floats,  are  made  with  corresponding  tenons,  which  project  through  the  crowns, 
and  are  riveted  on  the  bottom  of  the  lower  crown,  and  on  the  top  of  the  upper  crown ; 
this  construction  gives  the  requisite  strength  and  firmness,  with  buckets  of  much  thinner 
iron  than  was  necessary  under  any  of  the  old  arrangements ;   it  also  leaves  the  passages 
through  the  wheel  entirely  free  from  injurious  obstructions. 

15.  Mr.  Boy  den  has  also  designed  a  large  number  of  turbines  for  different  man- 
ufacturing establishments  in  New  England,  many  of  them  under   contracts  similar  to 
that  with  the  Appleton  Company,  and  has  accumulated  a  vast  number  of  valuable  ex- 
periments and  observations  upon  them,  which,  it  is  to  be  hoped,  he  will  find  time  to 
prepare  for  publication ;    as  such  opportunities  but  rarely  occur  to  engineers  so  able 
to  profit  by  them. 

16.  In  the  year  1849,  the  Manufacturing  Companies  at  Lowell  purchased  of  Mr. 
Boyden,  the  right  to  use  all  his  improvements  relating  to  turbines  and  other  hydraulic 
motors.     Since  that  time  it  has  devolved  upon  the  author,  as  the  chief  engineer  of  these 
companies,  to  design  and  superintend  the  construction  of  such  turbines  as  might  be 
wanted   for  their  manufactories,  and  to  aid   him  in   this  important  undertaking,  Mr. 
Boyden  has  communicated  to  him  copies  of  many  of  his  designs  for  turbines,  together 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE.  7 

with  the  results  of  experiments  upon  a  portion  of  them ;  he  has  communicated,  how- 
ever, but  little  theoretical  information,  and  the  author  has  been  guided,  principally,  by 
a  comparison  of  the  most  successful  designs,  and  such  light  as  he  could  obtain  from 
writers  on  this  intricate  subject. 

17.  The  first  designs,  prepared  by  the  author,  after  the  arrangement  with  Mr. 
Boyden  was  entered  into,  were  for  four  turbines  of  essentially  the  same  dimensions; 
namely,  two  for  the  Suffolk  Manufacturing  Company,  and  two  for  the  Tremont  Mills, 
for  the  purpose  of  furnishing  power  for  the  cotton-mills  of  these  companies  at  Lowell. 
These  turbines  'were  constructed  at  the  Lowell  Machine  Shop,  and  were  completed  in 
January,  1851. 

For  the  purpose,  principally,  of  estimating  the  success  of  these  turbines,  one  of 
them  was  fitted  up  with  a  complete  apparatus  for  measuring  its  power,  and  gauging 
the  quantity  of  water  discharged ;  the  gauging  apparatus  was  afterwards  used  to  make 
the  experiments  on  the  discharge  of  water  over  weirs  of  different  proportions,  for  the 
purpose  of  determining,  practically,  some  of  the  relations  required  to  be  known,  in 
order  to  compute  the  flow  of  water  through  such  apertures. 


DESCRIPTION  OF  THE  TUUBINE. 

18.  The  water  is  conducted  from  the  principal  feeder  to  the  mills  at  Lowell, 
called  the  Northern  Canal,  by  an  arched  canal,  or  penstock,  about  ninety  feet  in  length. 
The  forebay,  inside  the  wheel-house,  is  constructed  of  masonry,  and  has  a  general  width 
of  twenty  feet,  and  a  depth  of  water  of  fourteen  feet ;  the  channels  through  which  the 
water  passes,  are  so  capacious,  that  the  loss  of  fall  in  passing  from  the  Northern  Canal 
to  the  forebay,  is  scarcely  sensible.  During  the  experiments,  however,  the  head  of  the 
penstock  was  partially  closed  by  gates,  so  that  there  was  a  sensible  fall  at  that  time. 

The  entrance  of  the  arched  canal  is  protected  by  a  coarse  rack,  or  grating,  for 
the  purpose  of  preventing  large  floating  substances  from  entering  the  forebay ;  each 
turbine  is  also  separately  guarded  by  a  fine  rack,  placed  in  the  forebay,  which  prevents 
the  entrance  into  the  turbine  of  all  floating  substances  that  might  be  injurious.  Both 
racks  are  made  of  large  extent,  to  avoid  sensible  loss  of  head  to  the  water  in  passing 
through  them. 

The  extreme  rigor  of  the  New  England  winter  renders  it  necessary  to  afford  to 
water-wheels  of  all  descriptions,  complete  protection  from  the  cold.  The  result  is,  thai 
less  interraption  from  frost  is  experienced,  than  in  many  milder  climates.  The  wheel- 
house,  in  which  these  turbines  are  placed,  is  a  substantial  brick  building,  well  warmed 
in  the  winter  by  steam. 


g  EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 

After  passing  the  turbines,  the  water  is  conducted  by  an  arched  canal,  or  raceway, 
about  nine  hundred  feet  in  length,  to  the  lower  level  of  the  Western  Canal,  which 
serves  as  a  feeder  to  the  Mills  of  the  Lawrence  Manufacturing  Company. 

19.  Plate  I.  is  a  vertical  section  through  the  centre  of  the  turbine,  and  the  axis" 
of  the  supply  pipe. 

Plate  II.  is  a  plan  of  the  turbine,  and  wheelpit. 

Plate  III.,  Figure  1,  is  a  plan  of  nearly  one  fourth  part  of  the  disc  and  wheel. 
Figure  2  is  a  plan  of  the  whole  wheel,  the  guides,  and  garniture.  Figure  3  is  a  ver- 
tical section  through  both  crowns  of  the  wheel. 

The  same  letters  indicate  the  same  parts,  in  all  these  three  plates. 

20.  A,  the  forcbay,  in  which  the  level  of  the  water  is  nearly  the  same  as  in  the 
Northern  Canal ;   it  is  represented  at  the  usual  working  height. 

21.  JB,  the  surface  of  the  water  in  the  wheelpit,  represented  at  the  lowest  height  at 
which  the  turbine  is  intended  to  operate. 

22.  C,  the  masonry  of  the  ivheelpit.      The  faces  towards  the  wheel,  are  of  granite 
ashlar  work,  in  blocks  containing,  generally,  from  ten  to  forty  cubic  feet.     The  backing 
is  of  hard  mica  slate.     The  capping  course,  shown  particularly  on  Plate  II.,  is  neatly 
dressed  on  its  upper  surface.     The  whole  is  compactly  laid  in  hydraulic  cement. 

23.  D,  the  floor  of  the  wheelpit.     This  floor  sustains  the  weight  of  part  of  the  sup- 
ply pipe,  and  of  part  of  the  water  in  it,  and  all  the  rest  of  the  apparatus,  excepting 
the  wheel  itself  and  the  vertical  shaft,  which  are  supported  by  beams  and  braces,  directly 
from  the  side  walls  of  the  wheelpit.     It  was  necessary  that  the  floor  should  have  suffi- 
cient stiffness  to  resist  the  great  upward  pressure  which  takes  place  when  the  wheelpit 
is  kept  dry  by  pumps,  in  order  to  permit  repairs  to  be  made.     The  walls  of  the  wheel- 
pit  are  built  upon  the  floor;  —  there  was,  consequently,  no  danger  of  the  whole  floor 
being  pressed  upwards,  but  the  great  width  of  the  pit,  (twenty-four  feet,)  would  allow  the 
floor  to  yield  in  the  centre,  unless  it  had  great  stiffness. 

To  meet  these  requirements,  three  cast-iron  beams  are  placed  across  the  pit,  the 
ends  extending  about  a  foot  under  the  walls,  on  each  side ;  on  these  are  laid  thick 
planks  which  are  firmly  secured  to  the  castriron  beams,  by  bolts.  To  protect  the  thick 
planking  from  being  worn  out  by  the  constant  action  of  the  water,  they  are  covered 
with  a  flooring  of  one  inch  boards,  which  can  be  easily  renewed  when  necessary. 

24.  E,  the  wrought  iron  supply  pipe.     This  is  constructed  of  plate  iron,  f  inch  thick, 
riveted  together  in  a  similar  manner  to  steam  boilers.     The  horizontal  part  is  nine  feet 
in  diameter,  the  curved  part  gradually  diminishes  in  diameter,  to  its  junction  with  the 
upper  curb.     The  upper  end  of  the  supply  pipe  is  terminated  by  a  cast-iron  ring  F, 
turned  smooth  on  the  face,  to  receive  the  wooden  head  gate.     The  supply  pipe  is  also 
furnished  with  the  man  hole  and  ventilating  pipe  G,  and  the  leak  box  H.     The  use  of 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE.  9 

the  latter  is,  to  catch  the  leakage  of  the  head  gate,  whenever  it  is  closed  for  repairs 
of  the  wheel;  at  such  times,  the  leakage  is  carried  off  into  the  raceway,  below  the 
wheelpit,  by  a  six  inch  pipe,  furnished  with  a  valve  which  can  be  opened  and  shut  at 
pleasure. 

25.  I,   the  cast-iron  curls.     These  conduct  the  water  from  the  wrought  iron  sup- 
ply pipe,  to  the  disc  K.     The  curbs  are  made  in  four  parts,  for  the  convenience  of  the 
founder.     The  surfaces  at  which  they  are  joined,  are  turned  true  in  a  lathe,  packed  with 
red  lead,  and  bolted  together  with  bolts  one  and  a  half  inches  diameter,  placed  about 
six  inches  apart.     The  general  thickness  of  the  iron  is  one  and  a  quarter  inches.     The 
flanges  are  two  inches  thick.     The  upper  curb  has  a  projection  cast  on  it,  to  receive  the 
disc  pipe.     The  lower  curb  is  finished  on  all  sides ;  the  outside,  to  permit  the  regulating 
gate  to  be  moved  up  and  down  easily ;  the  inside,  to  present  a  smooth  surface  to  the 
water,  and  to  match  accurately  with  the  garniture  L. 

The  curbs  are  supported  from  the  wheelpit  floor  by  four  columns,  two  of  which 
are  shown  at  N '  N,  plate  L,  resting  on  the  cast-iron  beam  0;  this  is  placed  on  the 
floor,  for  the  purpose  of  distributing  the  weight.  The  centres  of  the  columns  are 
thirteen  inches  from  the  outside  circumference  of  the  wheel.  The  beams  N'  rest 
immediately  upon  the  columns,  and  the  curb  upon  the  beams,  the  latter  projecting 
over  the  columns  far  enough  for  that  purpose.  The  beams  N'  also  act  as  braces 
from  the  wheelpit  wall  to  the  curb,  and  are  strongly  bolted  at  each  end. 

26.  K,  the  disc.     This  is  of  cast-iron,  one  and  a  half  inches  thick,  and  is  turned 
smooth  on  the  upper  surface,  and  also  on  its   circumference.      It  is  suspended  from 
the   upper' curb,  by  means  of  the  disc   pipes   MM.     The   disc   carries   on   its  upper 
surface  thirty-three   guides,  or  leading  curves,  for  the   purpose  of  giving   the  water, 
entering   the   wheel,  proper   directions.      They   are    made   of  Kussian   plate   iron,  one 
tenth  of  an  inch  in  thickness,  secured  to  the  disc  by  tenons,  passing  through  corre- 
sponding  mortices,  cut   through   the    disc,   and   are   riveted   on    the    under-side.      The 
upper  corners  of  the  guides,  near  the  wheel,  are  connected  by  the  garniture  L,  which 
is  intended  to  diminish  the  contraction  of  the  streams   entering  the  wheel,  when  the 
regulating  gate  is  fully  raised.     The  garniture  is  composed  of  thirty-three    pieces  of 
cast-iron,  or  one  to  fill  each  space  between  the  guides;   these  pieces  of  cast-iron  are, 
necessarily,   of  irregular  form ;    for  a  top   view   of  them   see   L,  plate  III.,  figure  2. 
They  are   also   shown   in   section   at  plate  I.      They  are   carefully  fitted   to   fill   the 
spaces  between  the  guides ;  above  the  top  of  the  guides,  the  adjoining  pieces  are  in 
contact ;    they  are   strongly  riveted    to    the   guides,   and   to   each   other.      After   they 
were  all  fitted  and  riveted,  the  disc  was  put  in  a  lathe,  and  the  top,  the  periphery, 
and  a  part  of  the  inside  of  the  garniture,  were  turned  off,  so  that  it  would  fit  accu- 


10  EXPERIMENTS  UPON  THE  TBEMONT   TURBINE. 

rately,  but  easily,  to  the  corresponding  part  of  the  lower  curb.     The  disc  is  not  fast- 
ened  to  the  lower  curb,  but  is  retained  in  its  place,  horizontally,  by  the  latter. 

27.  MM,  the  disc  pipe.      The   disc  is  fastened   to   the  bottom   of  the   disc   pip« 
by  fifteen  tap  screws,  one  and  a  quarter  inches  in  diameter.     As  there  is  a  vertical 
pressure   on  the  disc,  due  to  the  pressure  of  the  whole  head,  on  its  horizontal  area, 
the  disc  pipe  and  its  fastenings  require  to  be  very  strong.     The  pipe  is  eight  and  a  half 
inches  diameter,  inside,  or  one  and  a  half  inches  larger  than  the  shaft  passing  through 
it,  and  is  one  and  a  quarter  inches  thick.     The  upper  flange  is  furnished  with  adjusting 
screws,  by  which  the  weight  is  supported  upon  the  upper  curb,  and  which  afford  the 
means  of  adjusting  the  height  of  the  disc.     The  escape  of  water  between  the  upper 
curb  and  the  upper  flange  of  the  disc  pipe,  is  prevented  by  a  band  of  leather  on  the 
outside,  which  is  retained  in  its  place  by  the  wrought  iron  ring  P.     This  ring  is  made 
in   two   segments.     The   top  of  the  disc   pipe,  just   below   the    upper   flange,  has  two 
projections,  or   wings,  which   fit   into  corresponding  recesses  in  the  top  of  the  curb; 
these  are  to  prevent  the  disc  from  rotating  in  the  opposite    direction  to  the  wheel, 
to  which  there  is  a  powerful  tendency,  arising  from  the  reaction  of  the  water  issuing 
from  the  guides. 

28.  R  R,  the  regulating  gate.      This  is  represented  on  the  section,  at  plate  I.,  as 
fully  raised,  and  in  this  position  the  wheel  would  be  giving  its  full  power.     The  gate 
is  of  cast-iron,  the  cylindrical  part  is  one  inch  thick,  the   upper  part  of  the  cylinder 
is  stiffened  by  a  rib,  to  which  are  attached  three  brackets,  one  of  which  is  shown  at 
S,  plate   I.,  and    the   two   others   at  S  S,  plate  II.      To  these  brackets   are  attached 
wrought  iron  rods,  by  which  the  gate  is  raised  and  lowered.     The  brackets  are  attached 
to  the  gate  at  equal  distances,  and  therefore  the  rods  support  equal  parts  of  its  weight. 
To  one  of  the  rods  is  attached  the  rack  V.     The  other  two  rods  are  attached,  by  means 
of  links,  to  the  levers  T  T,  plate  II.     The  other  ends  of  these  levers  carry  geared  arch 
heads,  into  which,  and  into  the  rack  V,  work  three  pinions,  W,  of  equal  pitch  and  size, 
fastened  to  the  same  shaft.     As  the  fulcrums  of  the  levers  T  T,  plate  II.,  are  exactly 
in  the  middle,  between  the  pitch  lines  of  the  arch  heads  and  the  points  to  which  the 
rods  are  attached,  it  will  be  seen,  that  by  the  revolution  of  the  pinion  shaft,  the  gate 
must  be  moved  up  or  down,  equally  on  all  sides.      The    shaft  on  which  the  pinions 
are  fastened,  is  driven  by  the  worm  wheel  X,  plates  I.  and  II. ;  this  is  driven  by  the 
worm  0,  either  by  the  governor  Y,  or  the  hand  wheel  Z.     The  shaft  on  which  the 
worm  a  is  fastened,  is  furnished  with  movable  couplings,  which,  when  the  speed  gate 
is  at  any  intermediate  points  between  its  highest  and  lowest  positions,  are  retained 
in  place  by  spiral  springs ;  in  either  of  the  extreme  positions,  the  couplings  are  sep- 
arated by  means  of  a  lever,  moved  by  pins  in  the  rack   F;   by  this  means  both  the 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE.  H 

regulator  and  hand  wheel  are  prevented  from  moving  the  gate  in  one  direction,  when 
the  gate  has  attained  either  extreme  position.  If,  however,  the  regulator  or  hand 
wheel  should  be  moved  in  the  opposite  direction,  the  couplings  would  catch,  and  the 
gate  would  be  moved.  The  weight  of  the  gate  is  counterbalanced  by  weights  attached 
to  the  levers  T  T,  and  by  the  intervention  of  a  lever  to  the  rack  F;  by  this  arrange- 
ment, both  the  governor  and  hand  wheel  are  required  to  operate,  with  only  the  force 
necessary  to  overcome  the  friction  of  the  apparatus. 

29.  bb  The  wheel.  This  consists  of  a  central  plate  of  cast-iron,  and  of  two 
crowns,  cc,  of  the  same  material,  to  which  the  buckets  are  attached.  The  central 
plate  and  the  crowns  are  turned  accurately  in  a  lathe,  for  the  purpose  of  balancing 
them,  and  also  to  diminish,  as  much  as  possible,  the  resistances  in  moving  rapidly 
through  the  water.  The  lower  crown  is  fastened  to  the  central  plate,  as  shown  at 
figures  1  and  3,  plate  III.  These  figures  also  show,  at  cc,  the  form  of  the  crowns; 
the  upper  and  lower  crowns  are  precisely  alike ;  they  are  nine  and  a  half  inches 
wide.  At  the  inner  edge,  and  at  the  circumference,  the  thickness  is  0.625  inches, 
and  at  5.5  inches  from  the  inner  edge,  where  they  have  the  greatest  thickness,  they 
are  one  inch  thick. 

By  reference  to  figure  1,  plate  III,  it  will  be  seen  that  the  buckets  do  not 
extend  to  the  circumference  of  the  crowns.  In  the  direction  of  the  radius,  the  ends 
of  the  buckets  are  0.25  inches  from  the  circumference.  This  is  for  the  purpose  of 
permitting  the  wheel  to  be  handled  with  less  danger  of  injuring  the  ends  of  the  floats; 
as  these  are  filed  down  to  an  edge,  they  would  be  very  likely  to  be  damaged  during 
the  construction  of  the  wheel,  if  they  were  not  guarded  by  the  slight  projection  of 
the  crowns.  This  construction  also  enables  the  grooves  in  the  crowns  to  afford  more 
perfect  support  to  the  ends  of  the  buckets,  and  also  permits  a  tenon  to  be  nearly  at 
the  extremity  of  the  bucket. 

The  buckets  are  forty-four  in  number,  and  are  of  the  form  represented  on  plate 
III.,  figure  1.  They  are  made  of  plate  iron  of  excellent  quality,  imported  from  Russia 
for  the  purpose,  they  are  •£-%  of  an  inch  in  thickness,  and  are  secured  to  the  crowns 
in  the  following  manner. 

The  crowns  having  been  first  turned  to  the  required  form,  grooves  are  cut  in 
them  of  the  exact  form  of  the  buckets,  to  the  depth  e  e,  figure  3,  plate  III. ;  this  depth 
is  0.1  inches  at  the  edges  and  0.5  inches  near  the  middle.  These  grooves  are  cut 
in  a  machine  contrived  for  the  purpose,  in  which  the  cutting  tool  is  guided  by  a  cam. 
Three  mortices  for  each  bucket  are  then  cut  through  each  crown ;  corresponding 
tenons  are  left  on  the  buckets;  the  latter  are  bent  to  the  required  form,  by  means 
of  a  pair  of  dies,  prepared  for  the  punpose,  the  plate  iron  having  been  first  moderately 
heated.  The  tenons  of  all  the  buckets  are  then  entered  into  the  mortices  in  both 


12  EXPERIMENTS   UPON  THE   TREMONT   TURBINE. 

crowns,  the  latter  are  then  drawn  together,  by  means  of  a  number  of  screws  applied  to 
different  parts  of  the  circumference,  and  when  the  edges  of  the  buckets  are  drawn  into 
the  bottom  of  the  grooves,  the  tenons  are  riveted  on  the  opposite  sides.  This  construc- 
tion gives  great  stability  to  the  buckets,  and  permits  the  use  of  very  thin  iron.  - 

30.  dd  The  vertical  shaft.     This  is  of  wrought  iron,  and   is  accurately  turned 
in  every  part 

The  diameters  are  as  follows :  — 

Below  the  hub  of  the  wheel, 7   inches. 

In  the  hub  of  the  wheel, 7£  " 

Between  the  top  of  the  hub  and  the  lower  bearing,  7  " 
Between  the  bottom  of  the  lower  bearing  and  the 

hub  of  the  bevel  gear, .  8  " 

In  the  hub  of  the  bevel  gear, 8£  " 

From  the  top  of  the  hub  of  the  bevel  gear  to  the 

suspeusion  box, 8  " 

By  reference  to  plate  I.,  it  will  be  seen  that  the  shaft  does  not  run  upon  a  step 
at  the  bottom,  but  upon  a  series  of  collars,  resting  upon  corresponding  projections  in 
the  suspension  box  e'.  The  part  of  the  shaft  on  which  the  collars  are  placed,  is  made 
separate  from  the  main  shaft,  and  is  joined  to  it  at  /,  by  means  of  a  eocket  in  the  top 
of  the  main  shaft,  which  receives  a  corresponding  part  of  the  collar  piece.  The  collars 
are  made  of  cast  steel ;  they  are  separately  screwed  on,  and  keyed  to  a  wrought  iron 
spindle. 

31.  e'    The  suspension  lox.     This   is  made   in   two   parts,  to  admit   of  its  being 
taken  off,  and  put  on  the  shaft ;  it  is  lined  with  babbit  metal,  a  soft  composition  con- 
sisting principally  of  tin.     It  is  found  that  bearings  thus  lined  will  carry  from  fifty  to 
a  hundred  pounds  to  the  square  inch,  with  every  appearance  of  durability. 

32.  /'/',  The  upper  and  lower  bearings.     These  are  of  cast-iron,  lined  with  babbit 
metal ;  they  are  retained  in  position,  horizontally,  by  means  of  adjusting  screws ;  ver- 
tically, their  weight  is  sufficient.     The  parts  of  the  shaft  inside  the  hubs  of  the  wheel 
and  the  bevel  gear,  are  made  slightly  tapering,  about  -fa  of  an  inch  in  diameter  in 
the  length  of  the  hubs ;  the  hubs  are  bored  out  with  the  same  taper,  but  a  very  little 
smaller  in  diameter ;  they  are  then  drawn  on  by  a  powerful  screw  purchase,  and  in 
this  manner   are  made   to   fit  very  tight.      To  prevent   danger  of  bursting  the  hubs, 
they  are   before   being   drawn   on   or  bored  out,  strongly  hooped  with  wrought  iron 
hoops,  driven  on  hot. 

33.  The  suspension  box  e'  (art.  31,)  rests  upon  the  gimbal  g,  plates  I.  and  II.     The 
gimbal  itself  is  Supported  on  the  frame  hh,  by  adjusting  screws,  which  give  the  means 


EXPERIMENTS   UPON   THE   TREMONT   TURBINE.  Jg 

of  raising  and  lowering  the  suspension  box,  and,  with  it,  the  vertical  shaft  and  wheel. 
It  will  be  perceived,  by  the  arrangement  of  the  bearings  above  and  below  the  bevel 
gear,  that  no  lateral  strain  can  be  thrown  upon  the  suspension  box.  The  construction  of 
the  shaft  will  evidently  not  admit,  with  safety,  of  lateral  strain  at  the  suspension  box,  and 
it  is  accordingly  so  arranged  that  this  box  is  free  to  oscillate  horizontally  in  any  direction, 
a  small  quantity,  in  case  any  irregularity  in  the  form  of  the  shaft  should  require  it. 

The  lower  end  of  the  shaft  is  fitted  with  a  cast  steel  pin  i,  plate  I.  This  is  retained 
in  its  place  by  the  step,  which  is  made  in  three  parts,  and  lined  with  casehardened 
wrought  iron.  The  step  is  furnished  with  adjusting  screws,  by  means  of  which  the 
shaft  can  be  moved  horizontally  in  any  direction,  a  small  distance. 

The  weight  of  the  wheel,  upright  shaft,  and  bevel  gear,  is  supported  by  means 
of  the  suspension  box  e,'  on  the  frame  k,  which  rests  upon  the  long  beams  m,  reaching 
across  the  wheelpit,  and  supported  at  the  ends  by  the  masonry,  and  also  at  intermediate 
points  by  the  braces  nn. 

From  economical  considerations  the  dijfuser,  described  at  art.  12,  was  omitted  at 
the  Tremont  Turbine ;  a  large  majority  of  the  turbines  in  use  at  Lowell,  however,  are 
fitted  up  with  that  apparatus. 

34.  The  following  are  some  of  the  dimensions  of  the  turbine,  carefully  taken  after 
the  parts  were  finished :  — 

Height  between  the  upper  and  lower  crowns,  at  the  outer  extrem- 
ities of  the  buckets,  a  mean  of  44  measurements,  ....  0.9314  feet. 

Height  between  the  upper  and  lower  crowns,  at  the  inner  extrem- 
ities of  the  buckets,  a  mean  of  44  measurements,  ....  0.9368  " 

Height  between  the  crowns,  at  a  point  5.5  inches  from  the  inner 
edges  of  the  crowns,  (designed  to  be  0.75  inches  less  than 
at  the  inner  edges,)  . .' 0.8743  « 

Shortest  distance  between  the  outer  extremities  of  the  buckets 

and  the  next  adjacent  buckets,  a  mean  of  132  measurements,  0.18757  " 

Shortest  horizontal  distance  between  two  adjacent  guides,  taken 
at  the  top  of  the  circumferential  part  of  the  disc,  a  mean 
of  33  measurements, 0.1960  a 

Do.     do.     at  the  bottom  of  the  garniture, 0.2117       " 

Do.     do.     half-way  up  between  the  disc  and  the  garniture,     .     .     0.2044       " 

The  shortest  distance  between  the  guides,  by  a  mean  of  the 

whole  99  measurements, 0.20403  " 

Height  from  the  top  of  the  circumferential  part  of  the  disc  to  the 

bottom  of  the  garniture,  a  mean  of  33  measurements,  .  .  0.97090  " 


14  EXPERIMENTS   UPON   THE  TREMONT   TURBINE. 

35.     The  following  are  some  of  the  most  important  dimensions  of  the  apparatus ; 

they  are   taken   from   the    original   designs,  which   were   very  closely  followed  in  the 
construction. 

Diameter  of  the  exterior  circumference  of  the  crowns  of  the  wheel,  8.333  feet. 

"           "       outer  extremities  of  the  buckets, 8.292  " 

"  "       interior  edges  of  the  crowns,  and  inner   edges   of 

the  buckets, 6.750  « 

"           "       outside  of  the  cylindrical  part  of  the  regulating  gate,  6.729  " 

"           "       inside  of  the  cylindrical  part  of  the  regulating  gate,  6.562  " 
a           "       of  the  outside  of  the  lower  curb,  taken  below  the 

flange, 6.542  « 

"           "       inside  of  the  lower  curb,  taken  at  the  top,  .     .     .  6.333  a 
"          "       inside  of  the  lower  curb,  taken  at  the  top  of  the 

guides, 6.167  « 

«          «       lower  part  of  the  disc, 6.729  « 


DESCRIPTION  OF  THE  APPARATUS  USED  IN  THE  EXPERIMENTS  ON  THE  TREMONT  TURBINE. 

36.  The  details  of  this  apparatus  are  represented  on  plate  IV. 

The  useful  effect  was  measured  with  a  Prony  dynamometer,  represented  in  sectional 
elevation  at  figure  1,  and  in  plan  at  figure  2. 

37.  The  friction  pulley  A  is  of  cast-iron  5.5  feet  in  diameter,  two  feet  wide  on  the 
face,  and  three  inches  thick.     It  is  attached  to  the  vertical  shaft  by  the  spider  B,  the 
hub  of  which  occupies  the  place  on  the  shaft  intended  for  the  bevel  gear. 

The  friction  pulley  has,  cast  on  its  interior  circumference,  six  lugs,  C  C,  correspond- 
ing to  the  six  arms  of  the  spider.  The  bolt  holes  in  the  ends  of  the  arms  are  slightly 
elongated  in  the  direction  of  the  radius,  for  the  purpose  of  allowing  the  friction  pulley 
to  expand  a  little  as  it  becomes  heated,  without  throwing  much  strain  upon  the  spider. 
When  the  spider  and  friction  pulley  are  at  the  same  temperature,  the  ends  of  the  arms 
are  in  contact  with  the  friction  pulley.  The  friction  pulley  was  made  of  great  thick- 
ness for  two  reasons.  When  the  pulley  is  heated,  the  arms  cease  to  be  in  contact  with 
the  interior  circumference  of  the  pulley,  consequently  they  would  not  prevent  the 
pressure  of  the  brake  from  altering  the  form  of  the  pulley.  This  renders  great  stiff- 
ness necessary  in  the  pulley  itself.  Again,  it  is  found  that  a  heavy  friction  pulley 
insures  more  regularity  in  the  motion,  operating,  in  fact,  as  a  fly-wheel,  in  equalizing 
small  irregularities. 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE.  15 

38.  Tlic  brakes  E  and  F  are  of  maple  wood ;  the  two  parts  are  drawn  together  by 
the  wrought  iron  bolts  G  G,  which  are  two  inches  square. 

39.  T/te  bell  crank  F'  carries  at  one  end  the  scale  I,  and  at  the  other  the  piston 
of  the  hydraulic  regulator  K;  this  end  carries  also  the  pointer  L,  which  indicates  the 
level  of  the  horizontal  arm.     The  vertical  arm  is  connected  with  the  brake  F,  by  the 
link  M,  figure  3. 

40.  The  hydraulic  regulator  K,  figures  1,  2,  and   5,  is  a  very  important  addition 
to  the  Prony  dynamometer,  first  suggested   to  the  author   by  Mr.  Boyden  in    1844. 
Its  office  is  to  control  and  modify  the  violent  shocks  and  irregularities,  which  usually 
occur  in  the  action  of  this  valuable  instrument,  and  are  the  cause  of  some  uncertainty 
in  its  indications. 

The  hydraulic  regulator  used  in  these  experiments,  consisted  of  the  cast-iron 
cylinder  K,  about  1,5  feet  in  diameter,  with  a  bottom  of  plank,  which  was  strongly 
bolted  to  the  capping  stone  of  the  wheelpit,  as  represented  in  figure  1.  In  this  cylin- 
der, moves  the  piston  N,  formed  of  plate  iron  0.5  inches  thick,  which  is  connected  with 
the  horizontal  arm  of  the  bell  crank  by  the  piston  rod  0.  The  circumference  of  the 
piston  is  rounded  off,  and  its  diameter  is  about  ^  inch  less  than  the  diameter  of  the 
interior  of  the  cylinder.  The  action  of  the  hydraulic  regulator  is  as  follows.  The 
cylinder  should  be  nearly  filled  with  water,  or  other  heavy  inelastic  fluid.  In  case  of 
any  irregularity  in  the  force  of  the  wheel,  or  in  the  friction  of  the  brake,  the  tendency 
will  be,  either  to  raise  or  lower  the  weight ;  in  either  case  the  weight  cannot  move, 
except  with  a  corresponding  movement  of  the  piston.  In  consequence  of  the  inelas- 
ticity of  the  fluid,  the  piston  can  move  only  by  the  displacement  of  a  portion  of  the 
fluid,  which  must  evidently  pass  between  the  edge  of  the  piston  and  the  cylinder,  and 
the  area  of  this  space  being  very  small,  compared  to  the  area  of  the  piston,  the  motion 
of  the  latter '  must  be  slow ;  giving  time  to  alter  the  tension  of  the  brake  screws  before 
the  piston  has  moved  far.  It  is  plain  that  this  arrangement  must  arrest  all  violent 
shocks,  but,  however  violent  and  irregular  they  may  be,  it  is  evident  that,  if  the  mean 
force  of  them  is  greater  in  one  direction  than  in  the  other,  the  piston  must  move  in 
the  direction  of  the  preponderating  force,  the  resistance  to  a  slow  movement  being  very 
slight.  A  small  portion  of  the  useful  effect  of  the  turbine  must  be  expended  in  this 
instrument ;  probably  less,  however,  than  in  the  rude  shocks  the  brake  would  be  sub- 
ject to  without  its  use. 

41.  For  the  purpose  of  ascertaining  the   velocity  of  the  wheel,  a   counter   was 
attached  to  the  top  of  the  vertical  shaft,  so  arranged  that  a  bell  was  struck  at  the  end 
of  every  fifty  revolutions  of  the  wheel. 

42.  To  lubricate  the  friction  pulley,  and  at  the  same  time  to  keep  it  cool,  water 
was  let  on  to  its  surface  in  four  jets,  two  of  which  are  shown  in  figure  2,  plate  IV. 


IQ  EXPERIMENTS   UPON   THE   TREMONT   TURBINE. 

These  jets  were  supplied  from  a  large  cistern,  in  the  attic  of  the  neighboring  cotton- 
mill,  kept  full,  during  the  working  hours  of  the  mill,  by  force-pumps.  The  quantity  of 
water  discharged  by  the  four  jets  was,  by  a  mean  of  two  trials,  0.0288  cubic  feet  per 
second. 

In  many  of  the  experiments  with  heavy  weights,  and  consequently  slow  velocities, 
oil  was  used  to  lubricate  the  brake,  the  water,  during  the  experiment,  being  shut  off. 
It  is  found  that,  with  a  small  quantity  of  oil,  the  friction  between  the  brake  and  the 
pulley,  is  much  greater  than  when  the  usual  quantity  of  water  is  applied ;  consequently, 
the  requisite  tension  of  the  brake  screws  is  much  less  with  the  oil,  as  a  lubricator, 
than  with  water.  This  may  not  be  the  whole  cause  of  the  phenomenon,  but, 
whatever  it  may  be,  the  ease  of  regulating  in  slow  velocities  is  incomparably  greater 
with  oil  as  a  lubricator,  than  with  water  applied  in  a  quantity  sufficient  to  keep  the 
pulley  cool.  The  oil  was  allowed  to  flow  on  in  two  fine  continuous  streams;  —  it 
did  not,  however,  prevent  the  pulley  from  becoming  heated  sufficiently  to  decompose 
the  oil,  after  running  some  time,  which  was  distinctly  indicated  by  the  smoke  and 
peculiar  odor.  When  these  indications  became  very  apparent,  the  experiment  was 
stopped,  and  water  let  on  by  the  jets,  until  the  pulley  was  cooled.  As  the  pulley 
became  heated,  the  brake  screws  required  to  be  gradually  slackened. 

In  the  experiments,  in  table  II.,  the  lubricating  fluid  was  as  follows. 

In  the  first  twenty-six  experiments,  water  alone  was  used. 

In  the  four  experiments  numbered  from  27  to  30,  three  gallons  of  linseed  oil  were 
used. 

In  all  the  experiments  requiring  a  lubricator,  and  numbered  from  31  to  48, 
inclusive,  linseed  oil  was  used. 

In  experiments  49  and  50,  resin  oil  was  used. 

In  experiments  numbered  from  51  to  60,  inclusive,  water  alone  was  used. 

In  experiment  61,  resin  oil  was  used. 

In  experiment  62,  resin  oil  and  a  small  stream  of  water  were  used;  —  in  the 
latter  part  of  the  experiment,  a  good  deal  of  steam  was  generated  by  the  heat 
of  the  friction  pulley. 

In   experiment   63,   resin   oil   alone   was   used. 

In  experiments   numbered   from    66    to    72,  inclusive,  water   alone   was  used. 

In  experiments  numbered  from  73  to  79,  inclusive,  resin  oil  and  a  small  stream 
of  water  were  used. 

In   experiments   numbered   from    81   to    84,  inclusive,  water   alone    was   used. 

In   experiments   85   and    86,  resin   oil   and    a  small  stream  of  water  were  used. 

In   experiment    87,  resin    oil   alone   was   used. 

In   experiments  90  and   91,  water   alone   was   used. 


EXPERIMENTS    UPON   THE   TREMONT   TURBINE.  17 

In   experiment    92,  resin   oil   and   a   small   stream   of  water  were   used. 

43.  A   special   apparatus  was   provided    to    indicate    the  direction   in  which  the 
water  left  the  wheel.      For  this  purpose  the  vane  P,  figures  1,  6,  and  7,  plate  IV.. 
was  placed  near  the  circumference  of  the  wheel,  and  was  keyed  on  to  the  vertical  shaft 
Q,  which  turned  freely  on  a  step  resting  on   the  wheelpit  floor.     The  upper  end  of 
the  shaft  carried  the  hand  E,  figures  1  and  4,  and  directly  under  the  hand  Avas  placed 
the  graduated  semicircle  S,  divided  into  180°.     When  the  vane  was  parallel  to  a  tan- 
gent to  the  circumference  of  the  wheel,  drawn  through  the  point  nearest  to  the  axis 
of  the  vane,  and  the  vane  was  in  the  direction  of  the  motion  of  the  wheel,  the  hand 
pointed  at  0°,  and,  consequently,  wnen  the  vane  was  in  the  direction  of  the  radius  of 
the  wheel,  the  hand  pointed  at  90°.      To  prevent  sudden  vibrations  of  the  vane,  a 
modification  of  the  hydraulic  regulator  was  attached  to  the  lower  part  of  the  vane 
shaft     This  apparatus  is  represented  in  detail  by  figures  6  and  8. 

44.  The    quantity   of  water   discharged    by   the    wheel   was   gauged    at   a   weir 
erected  for  the  purpose  at  the  mouth  of  the  wheelpit.     It  is  represented  on  plate  V. 

Figure  1  is  a  plan,  and  figure  2  a  section,  showing  the  relative  positions  of  the 
turbine  A,  the  grating  B,  the  gauge  box  C,  and  the  two  divisions  or  bays  of  the  weir, 
D,  and  E. 

As  the  water  issued  from  the  orifices  of  the  turbine  with  considerable  force, 
particularly  when  the  velocity  of  the  wheel  was  much  quicker  or  slower  than  that 
corresponding  to  the  maximum  coefficient  of  effect,  there  were  often  such  violent 
commotions  in  the  wheelpit,  that,  \mless  some  mode  was  adopted  to  diminish  them 
before  the  water  reached  the  weir,  or  even  the  place  where  the  depths  on  the  weir 
were  measured,  it  would  have  been  impossible  to  make  a  satisfactory  gauge  of  the 
water.  For  this  purpose  the  grating  B,  figures  1  and  2,  was  placed  across  the  wheelpit. 
This  grating  presented  numerous  apertures,  nearly  uniformly  distributed  over  its  entire 
area,  through  which  the  water  must  pass.  In  the  experiments  with  a  full  gate,  the  fall 
from  the  upper  to  the  lower  side  of  the  grating  was  generally  from  three  to  four  inches. 
The  combined  effect  of  this  fall  and  of  the  numerous  small  apertures,  was,  to  obliterate 
almost  entirely  the  ,  whirls  and  commotions  of  the  water  above  the  grating.  About 
4.5  feet  in  length  of  the  grating  between  F  and  G,  figure  1,  was  so  nearly  closed, 
that  but  little  water  passed  through  that  part  of  the  grating;  —  this  made  it  very 
quiet  in  the  vicinity  of  the  gauge  box  C. 

Figure  3,  plate  V.,  is  an  elevation  of  the  weir.  The  two  bays  D  and  E  were 
of  nearly  equal  length,  —  the  crest  of  the  weir  was  almost  exactly  horizontal,  and 
the  extreme  variation  did  not  exceed  0.01  inch.  The  crest  of  the  weir  was  of  cast- 
iron,  planed  on  the  upper  edge  H,  figures  2  and  4,  and  also  on  the  upstream  face,  to  a 
point  1.125  inches  below  the  top;  —  below  this,  at  I,  figure  4,  there  was  a  small  bevel, 

3 


18  EXPERIMENTS   UPON  THE  TREMONT  TURBINE. 

also  planed,  the  slope  of  which,  on  an  average,  was  T8g  inch  in  a  height  of  f  inch ;  — 
the  remainder  of  the  casting  was  unplaned.  The  crest  of  the  weir  H  was  f  inch  thick, 
and  was  horizontal.  The  upstream  edge,  at  H,  was  a  sharp  corner.  The  ends  of  the 
weir  K,  figures  1,  2,  and  3,  were  of  wood,  and  of  the  same  form  as  the  crest  H,  except 
that  there  was  no  bevelled  part  corresponding  to  I,  figure  4.  The  crest  of  the  weir 
H  was  about  6.5  feet  above  the  floor  of  the  wheelpit.  The  ends  of  the  weir  K  pro- 
jected from  the  walls  of  the  wheelpit,  and  also  from  the  central  pier,  a  mean  distance 
of  1.235  feet.  The  length  of  the  bay  D,  was  8.489  feet,  and  of  the  bay  E,  8.491 
feet,  making  the  total  length  of  the  weir  16.98  feet. 

45.  The  depth  of  the  water  on  the  weir  was  taken  in  the  gauge  box  C,  figures 
1  and  2,  plate  V.,  by  means  of  the  hook  gauge  L,  which  is  represented  in  detail  by 
figures  9,  10,  and  11,  plate  IV. 

The  hook  gauge  is  the  invention  of  Mr.  Boyden,*  and  is  an  instrument  of  inesti- 
mable value  in  hydraulic  experiments.  All  other  known  methods  of  measuring  the 
heights  of  the  surface  of  still  water,  are  seriously  incommoded  by  the  effects  of 
capillary  attraction ;  this  instrument,  on  the  contrary,  owes  its  extraordinary  precision 
to  that  phenomenon.  At  figure  10,  plate  IV.,  the  point  of  the  hook  A,  is  represented 
as  coinciding  with  the  surface  of  the  water.  If  the  point  of  the  hook  should  be  a 
very  little  above  the  surface,  the  water  in  the  immediate  vicinity  of  the  hook,  would, 
by  capillary  attraction,  be  elevated  with  it,  causing  a  distortion  in  the  reflection  of  the 
light  from  the  surface  of  the  water.  The  most  convenient  method  of  observing  with 
this  instrument,  according  to  the  experience  of  the  author,  is,  first,  to  lower  the  point 
of  the  hook,  by  means  of  the  screw,  to  a  little  distance  below  the  surface ;  —  then  to 
raise  it  again  slowly  by  the  same  means,  until  the  distortion  of  the  reflection  begins 
to  show  itself,  —  then  to  make  a  slight  movement  of  the  screw  in  the  opposite  direction, 
so  as  just  to  cause  the  distortion  to  disappear ;  the  point  will  then  be  almost  exactly 
at  the  level  of  the  surface. 

With  no  particular  arrangements  for  directing  light  on  the  surface,  differences 
in  height  of  0.001  feet  are  very  distinct  quantities;  but  by  special  arrangements  for 
light  and  vision,  differences  of  0.0001  feet  might  be  easily  appreciated. 

As  this  instrument  cannot  be  efficiently  used  in  a  current,  it  was  placed  in  the 
box  C,  in  which  the  communication  with  the  exterior  was  maintained  by  the  hole  M, 


*  In  Versuche  iiber  den  avsflitss  des  wassers  dvrch  schieber,  hahne,  Happen  und  ventile,  by  Julius  Weisbach, 
Leipzig,  1842,  page  1,  is  described  an  instrument  for  observing  heights  of  water,  having  a  slight  resemblance 
to  the  hook  gauge;  it  was  however  used  by  Boyden  in  a  more  perfect  form,  several  years  previous  to  the 
publication  of  that  work. 


EXPERIMENTS   UPON   THE   TREMONT  TURBINE.  19 

when,  by  partially  obstructing  this  communication,  the  extent  of  the  oscillations  could 
be  diminished  at  will. 

For  the  most  perfect  observations,  it  is  essential  that  the  surface  of  the  water 
should  be  at  rest.  If,  however,  it  should  oscillate  a  little,  a  good  mean  may  be  obtained 
by  adjusting  the  point  of  the  hook  to  a  height  at  which  it  will  be  visible  above  the 
surface  of  the  water  only  half  the  time. 

The  movable  rod  to  which  the  hook  was  attached,  was  of  copper,  and  graduated 
to  hundredths  of  feet,  but,  by  means  of  the  vernier,  thousandths  were  measured,  and 
in  some  cases  ten  thousandths  were  estimated.  In  later,  and  more  perfect  forms  of 
this  instrument,  the  point  of  the  hook  is  immediately  under  the  graduation. 

46.  The  heights  of  the  water  in  the  forebay,  and  in  the  wheelpit,  were  taken 
by  means  of  gauges,  placed  in  the  gauge  boxes  p  and  q,  plate  II.     These  boxes  were 
similar  to  the  box    C,  plate  V.,  in  which  the  hook  gauge  was  placed.     Both  gauges 
were  graduated  to  feet  and  hundredths,  and  both  had  the  same  zero  point,  viz.,  the 
level  of  the  crest  of  the  weir,  so   that    the    difference    in   the   readings   at   the   two 
gauges,  gives,  at  once,  the  fall  acting  upon  the  wheel ;  and  the  difference  between  the 
depths  of  the  water  on  the  weir,  as  observed  at  the  hook  gauge,  and  the    reading 
at  the  gauge  q,  gives  the  fall  at  the  grating. 

In  consequence  of  want  of  space  in  plate  II.,  the  gauge  box  p  is  not  represented 
in  its  true  position,  —  it  was  actually  in  front  of  the  head  gate,  and  about  six  feet 
distant. 

47.  The  heights  of  the  regulating  gate  were  taken  at  the  rack  V,  plate  I.     The 
weights  used  for  measuring  the  useful  effect,  were  pieces  of  pig-iron  of  various  sizes, 
each  of  which  had   been  distinctly  marked  with  its  weight  by  Mr.  0.  A.  Richardson, 
the  official  sealer  of  weights  and  measures  for  the  City  of  Lowell. 


MODE  OF  CONDUCTING  THE  EXPERIMENTS. 

48.  A  separate  observer  was  appointed  to  note  each  class  of  data;  the  time 
of  each  observation  was  also  noted,  which  gave  the  means  of  identifying  simultaneous 
observations.  To  accomplish  this,  each  observer  was  furnished  with  a  watch  having 
a  second  hand ;  —  the  watch  by  which  the  speed  of  the  wheel  was  observed,  was  taken 
as  the  standard ;  all  the  others  were  frequently  compared  with  it,  and  when  the  vari- 
ations exceeded  ten  or  fifteen  seconds,  they  were  either  adjusted  to  the  standard,  or 
the  difference  noted. 

This  mode  of  observing  must,  evidently,  lead  to  more  precise  results  than  that  in 
which  a  single  observer,  however  skilful,  undertakes  to  note  all  the  phenomena,  or 


20 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 


even  several  of  them.  By  the  method  adopted,  a  regular  record  is  made  of  the 
state  of  things  at  very  short  intervals,  furnishing  the  data  for  a  mean  result  for  any 
required  period,  and  also  the  means  of  detecting,  in  most  cases,  the  causes  of  apparent 
discrepancies.  It  also  relieves  the  experimenter  from  the  distraction  of  having 
numerous  exact  observations  to  make  in  a  very  short  time,  and  leaves  him  much 
more  at  liberty  to  exercise  a  vigilant  watch  over  the  general  course  of  the 
experiment. 

49.  As  it  may  be  useful  to  experimenters,  not  accustomed  to  this  mode  of 
observing,  and,  at  the  same  time,  afford  the  reader  some  means  of  judging  of  the 
accuracy  of  the  results  obtained  in  these  experiments,  the  following  extracts  are 
given  from  the  original  note-books.  The  extracts  include  the  data  observed  for 
experiment  numbered  30  in  table  II.  This  experiment  is  selected,  simply,  because 
it  gave  the  maximum  coefficient  of  effect. 


WEIGHT  IN  THE  SCALE. 
Extract  from  the  note-look  of  the   author,  who  superintended  the   experiments. 

1,498  Ibs.  10i  oz. 


4h,  43',  added 

Weight  for  the  next  experiment, 


26 


1,524  Ibs.  lOfoz. 


SPEED  OF  THE  WHEEL. 
Extract  from   the  note-book  of  Mr.    Charles  Leonard. 


Times  at  which  the 
bell  struck. 

Differences. 

Times  at  which  the 
bell  struck. 

Differences. 

Times  at  which  the 
bell  struck. 

Differences. 

H. 

inin. 

sec. 

Seconds. 

H. 

min. 

sec. 

Seconds. 

H. 

min. 

sec. 

Seconds. 

4 

55. 

58.00 

5. 

0. 

52.00 

59.00 

5. 

4. 

47.00 

59.00 

56. 

56.50 

58.50 

1. 

50.75 

58.75 

5. 

45.50 

58.50 

57. 

55.25 

58.75 

2. 

49.50 

58.75 

6. 

44.25 

58.75 

58. 

54.25 

59.00 

3. 

48.00 

58.50 

7. 

43.00 

58.75 

59. 

53.00 

58.75 

NOT&    The  bell  struck  once  in  every  fifty  revolutions  of  the  wheel. 


EXPERIMENTS  UPON  THE  TBEMONT  TURBINE. 


21 


ELEVATION  OF  THE  POINTER  ON  THE  BELL   CRANK. 


Time. 

Height  of 
pointer, 
in  feet. 

Time. 

Height  of 
pointer, 
in  feet. 

Time. 

Height  of 
pointer, 
in  feet. 

H. 

min. 

sec. 

H. 

min. 

sec. 

H. 

min. 

sec. 

4. 

55. 

0. 

0.19 

4. 

59. 

30. 

0.20 

5. 

4. 

0. 

0.17 

30. 

0.13 

5. 

0. 

0. 

0.18 

30. 

0.18 

56. 

0.13 

30. 

0.19 

5. 

0.24 

30. 

0.14 

1. 

0.21 

30. 

0.18 

57. 

0.15 

30. 

0.17 

6. 

0.19 

30. 

0.19 

2. 

0.20 

30. 

0.19 

58. 

0.20 

30. 

0.19 

7. 

0.16 

30. 

0.19 

3. 

0.19 

30. 

0.14 

59. 

0.21 

30. 

0.19 

NOTE.      The    extremity  of  the    pointer  was  6.5  feet  from  the   fulcrum  of  the  bell    crank, 
horizontal  arms  of  the  bell  crank  were  level,  the  height  of  the  pointer  was  0.20  feet. 


When  the 


HEIGHT  OF  THE  WATER  ABOVE  THE  WHEEL. 
Taken   in   the  forebay   by  Mr.   John  Newell. 


Time. 

Height, 

in  feet. 

Time. 

Height, 
in  feet. 

Time. 

Height, 
in  feet. 

H. 

min.   |       sec. 

H. 

min. 

sec. 

H. 

min. 

sec. 

4. 

55. 

0. 

15.100 

4. 

59. 

30. 

15.110 

5. 

4. 

0. 

15.120 

30. 

15.100 

'  5. 

0. 

15.115 

30. 

15.120 

56. 

15.100 

30. 

15.120 

5. 

15.120 

30. 

15.100 

1. 

15.120 

30. 

15.115 

57. 

15.110 

30. 

15.110 

6. 

15.115 

30. 

15.115 

2. 

15.105 

30. 

15.110 

58. 

15.110 

30. 

15.100 

7. 

15.110 

30. 

15.100 

3. 

15.115 

30. 

15.110 

59. 

15.105 

30. 

15.125 

NOTE.    The  top  of  the  weir  is  the  zero  point  of  the  gauge  in  the  forebay. 


HEIGHT  OF  TrfE  WATER  AFTER  PASSING  THE   WHEEL. 
Taken  in  the  wheelpit   by  Mr.   Lloyd  Hixon. 


Time. 

Height, 

Time. 

Height, 

Time. 

Height, 

in  feet. 

in  feet. 

in  feet. 

H. 

min. 

sec. 

H. 

min. 

sec. 

H. 

min. 

sec. 

4. 

56. 

0. 

2.20 

5. 

0. 

0. 

2.21 

5. 

4. 

0. 

2.22 

• 

30. 

2.21 

30. 

2.21 

30. 

2.21 

57. 

2.21 

1. 

2.21 

5. 

2.21 

30. 

2.21 

30. 

2.21 

30. 

2.21 

58. 

2.21 

2. 

2.21 

6. 

2.21 

30. 

2.21 

30. 

2.21 

30. 

2.20 

59. 

2.20 

3. 

2.20 

7. 

2.22 

30. 

2.21 

30. 

2.20 

30. 

2.20 

NOTE.    The  top  of  the  weir  is  the  zero  point  of  the  gauge  in  the  wheelpit. 


22 


EXPERIMENTS   UPON   THE   TREMONT   TURBINE. 


HEIGHTS  OF  THE  WATER  ABOVE  THE  WEIR  BY  THE  HOOK  GAUGE. 
Observed  by   Mr.   Daniel   Haejffely. 


Time. 

Height, 

Time. 

Height, 

Time. 

Height, 

in  feet. 

in  feet. 

in  feet. 

H. 

mill. 

sec. 

H. 

min. 

sec. 

H. 

min. 

sec. 

4. 

57. 

5. 

1.8710 

5. 

i. 

10. 

1.8690 

5. 

4. 

35. 

1.8730 

58. 

15. 

1.8710 

i. 

45. 

1.8700 

0. 

50. 

1.8725 

58. 

50. 

1.8720 

2. 

15. 

1.8720 

6. 

25. 

1.8725 

59. 

20. 

1.8730 

2. 

50. 

1.8720 

6. 

55. 

1.8725 

59. 

50. 

1.8715 

3. 

15. 

1.8715 

7. 

20. 

1.8720 

5. 

0. 

15. 

1.8715 

3. 

40. 

1.8715 

7. 

45. 

1.8715 

0. 

45. 

1.8705 

4. 

5. 

1.8730 

I 

NOTE.     The  zero  of  the  hook  gauge  was  0.002  feet  below  the  top  of  the  weir. 

DIRECTION  OF  THE  WATER  LEAVING  THE  WHEEL. 
Observed  at  the  vane  index   by  Mr.   John    C.    Woodward. 


Time. 

Direction. 

Time. 

Direction. 

Time. 

Direction. 

H. 

min. 

sec. 

deg.  |  min. 

H. 

min. 

sec. 

deg. 

min. 

H. 

min. 

sec. 

deg. 

min. 

4. 

57. 

0. 

59. 

0. 

5. 

1. 

0. 

57. 

0. 

5. 

5. 

0. 

58. 

0. 

30. 

57. 

0. 

30. 

59. 

30. 

30. 

59. 

30. 

58. 

59. 

0. 

2. 

58. 

0. 

6. 

59. 

30. 

30. 

58. 

0. 

30. 

57. 

0. 

30. 

57. 

0. 

59. 

58. 

0. 

3. 

60. 

0. 

7. 

59. 

0. 

30. 

58. 

30. 

30. 

58. 

0. 

30. 

57. 

30. 

5. 

0. 

57. 

0. 

4. 

59. 

0. 

8. 

59. 

0. 

30. 

57. 

30. 

30. 

56. 

0. 

NOTE.      When  the   vane   pointed   in   the   direction   of   the   radius   of    the   wheel,   the   reading  of  the 
index  was  90°.      0°  was  in  the  direction  of  the  motion  of  the  wheel. 


50.  Previously  to  the  commencement  of  the  experiments,  the  apparatus  for 
measuring  the  useful  effect  was  carefully  adjusted.  The  bell  crank  was  balanced 
when  there  were  no  weights  in  the  scale.  For  this  purpose  the  link  M,  figure 
3,  plate  IV.,  was  removed,  and  the  chamber  of  the  hydraulic  regulator  filled  with 
water ;  —  weights  were  then  applied  to  the  top  of  the  bell  crank,  near  the  end 
to  which  the  hydraulic  regulator  was  attached,  until  the  whole  was  in  equilibrium; 
—  the  final  adjustment  was  made,  by  placing  a  weight  of  about  two  pounds  at 
the  extremity  of  one  of  the  horizontal  arms  of  the  bell  crank,  —  the  arm  was 
retained  horizontally  until  a  signal  was  given,  when  it  was  left  at  liberty  to 
descend,  and  the  time  occupied  in  descending  a  certain  distance  was  noted ;  — 
the  weight  was  then  removed  to  the  extremity  of  the  other  arm,  and  the  same 
process  repeated.  The  balance  weights  were  altered  until  the  times  of  descent 


EXPERIMENTS   UPON   THE   TREMONT   TURBINE.  23 

were  equal.  To  overcome,  as  much  as  possible,  the  friction  of  the  fulcrum,  the 
pin  forming  it  was  lubricated  with  sperm  oil,  and,  during  the  descent,  the  head 
of  the  pin  was  struck  lightly  and  rapidly  with  a  small  hammer. 

After  the  bell  crank  was  satisfactorily  balanced,  the  link  M  was  reattached, 
and  the  brake  adjusted,  by  means  of  the  screw  which  formed  the  connection 
between  the  link  and  the  brake.  It  was  adjusted  so  that  a  line  upon  the  brake 
was  perpendicular  to  the  axis  of  the  link,  when  the  horizontal  arm  of  the  bell 
crank  was  horizontal.  The  length  of  the  brake  was  then  measured  upon  this  line. 

The  length  of  the  brake  as  thus  measured  was  found  to  be     .     .     9.745     feet. 
The  effective  length    of  the  vertical   arm   of  the  bell  crank  was      4.500        " 
And   the    effective   length   of  the   horizontal   arm    to   which   the 

scale  was   hung,  was 5.000        " 

Consequently,  the    effective   length   of  the  brake  was    9-74^X  5  _  ^327773  " 

51.  The    gauges   in   the   forebay,  and   in   the   wheelpit,  were    carefully  adjusted 
by  levelling  from   the   top   of  the   weir.      This   was  repeated    by   different   persons, 
so  as  to  remove  all  chance  of  error. 

52.  The   hook    gauge    was    compared   with    the   weir,   by   a   different    method. 
When   the    regulating   gate    of  the    turbine   was   shut   down   as   tight   as   possible,  it 
was   still   found   that   a    quantity    of  water    leaked    into   the   wheelpit,   exceeding,   a 
little,  the    quantity  that   leaked    out   of  the    wheelpit,  so  that   a   small    quantity  con- 
tinued  to    run    over   the    weir.     The   principal   leak   into   the   wheelpit  was  between 
the    regulating   gate    and   the   lower   curb,   the   leather   packing   not   being   perfectly 
adjusted.     The   hook   gauge    was   firmly   attached   to    a   post,  placed   in   the  wheelpit 
for   that   purpose,    and    at    a   height    known   to    be    nearly   correct.      The   regulating 
gate    was   closed,  and   after   the    water   had   arrived   at   a   uniform   state,   the   height 
of  the   water   at   the   hook   gauge  was  noted,  and,  at   the    same  time,  the  depths  of 
the   water   on   the   weir   were    measured    directly   with    a   graduated   rule.      To   per- 
form  this   accurately,  a   board,  about  four  inches   long,  was   held   by  an   assistant  on 
the   crest   of  the   weir,  at   the    place  where"    it  was  intended  to  measure  the  depth ; 
—  the   author   then   applied   the   rule,   previously   well   dried,  vertically,  on   the    top 
of  the   weir,   in   front   of   the   board.      On   first   immersing   the   rule,   the   water  in 
contact  with   it   did   not   stand   at  the   true  level  of  the  surface,  but  formed  a  little 
hollow   around   the   rule ;     it   immediately    commenced   rising,   however,   and    after  a 
few  moments   came   to   a  level,   which   was   indicated   by   the   reflection   of  a   light 
from  the   surface,  a  lamp   being   held   by  an  assistant,  in  a  proper   position,  for  that 
purpose. 


24 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 


The   depths   on   the   weir,  tciken   in   the   manner  just   described,  February  20th, 
1851,  were  as  follows. 


Depths 
bay 

on  the  westerly 
of  the  weir. 

Inches. 

Depths  on  the  easterly 
bay  of  the  weir. 

Inches. 

0.37 
0.36 
0.37 
0.37 

0.36 
0.36 

0.36 
0.36 

Means    .     . 

0.3675 

....    0.36 

Or 

in  feet  . 

0.0306 

....     0.0300 

While  the  heights  given  in  the  preceding  table  were  being  measured,  the 
depth  by  the  hook  gauge  was  constantly  0.0318  feet;  consequently,  by  this  com- 
parison, the  zero  of  the  hook  gauge  was  0.0012  feet  below  the  mean  height  of 
the  top  of  the  weir,  in  the  westerly  bay,  and  0.0018  feet  below  the  mean  height 
in  the  easterly  bay,  or  0.0015  feet  below  the  mean  height  in  both  bays.  A 
similar  comparison  was  made  February  22d,  1851,  when  the  zero  of  the  hook 
gauge  was  found  to  be  0.0024  feet  below  the  mean  height  of  the  weir.  The 
mean  of  the  two  comparisons,  or  0.0020  was  adopted  as  the  correction  to  be  sub- 
tracted from  the  reading  of  the  hook  gauge,  to  give  the  mean  depth  upon  the 
weir. 

53.  During  the  experiments,  the  levels  of  the  water  in  the  upper  and  lower 
canals,  were  maintained  nearly  uniform.  The  height  of  the  lower  canal,  at  the 
place  where  the  water,  passing  the  weir,  fell  into  it,  varied  a  little,  depending 
upon  the  quantity  of  water  discharged  by  the  wheel.  It  was  highest  when  the 
wheel  was  running  with  the  regulating  gate  fully  raised,  and  the  brake  removed ; 
under  these  circumstances  the  surface  of  the  water  was  from  0.3  feet  to  0.4  feet 
below  the  top  of  the  weir.  In  the  other  experiments  with  the  regulating  gate 
fully  raised,  the  fall  from  the  top  of  the  weir  to  the  surface  of  the  water  in 
the  lower  canal,  was  from  0.4  feet  to  0.6  feet.  .  The  brackets  N  and  the  planks 
0,  figure  2,  plate  V.,  were  not  put  on  until  after  the  turbine  experiments  were 
concluded,  so  that  the  water  passing  the  weir,  met  with  no  obstruction  until  it 
struck  the  water  in  the  lower  canal. 

It  will  be  seen  by  the  experiments  on  the  weir,  (art.  127,)  that  the  obstruction, 
caused  by  the  planks,  was  scarcely  appreciable,  which  renders  it  certain  that  the 
effect  of  the  lower  canal,  in  obstructing  the  flow  over  the  weir,  must  have  been 
entirely  inappreciable. 


EXPERIMENTS   UPON  THE   TREMONT  TURBINE.  25 


DESCRIPTION  OF  TABLE  II.,    CONTAINING  THE    EXPERIMENTS    TJPON  THE  TURBINE  AT  THE 

TREMONT  MILLS. 

54.  The    data    obtained    by   direct   observation,   and    the    results    deduced   from 
them  by  calculation,  are  arranged  together,  for  convenience  of  reference,  in  table  II 

The  columns  numbered  ],  2,  and  3,  require  no  further  explanations  than  are 
contained  in  the  several  headings. 

55.  COLUMN  4.     Height  of  the  regulating  gate.      The    three   first   experiments   were 
made    under   circumstances   identical   in   every  thing,  except   that  the    height   of  the 
regulating  gate    was   varied   a   little,  for  the  purpose   of  ascertaining  the  height  giv- 
ing  the    maximum   coefficient    of  effect.      The   mean  height  between  the  crowns  of 
the   wheel,   at   the   inner   edges   of  the   buckets,  was  0.9368  feet,  or  11.2416  inches; 
the    curvature  of  the  disc  and  garniture,  however,  rendered  it-  necessary  to   raise  the 
gate  rather  more    than    this,  in    order   to   present   the   most  favorable  aperture.      By 
a  comparison  of  the  first  three  experiments,  it  appears  that  the  most  favorable  result 
was  obtained,  with   the    gate  raised    to   a   height   of    11.50   inches,    or   a    little   less; 
the    succeeding  experiments,   numbered  from  4   to  50,  inclusive,  were  made  with  the 
regulating    gate  raised    to  the  full   height,  or  to  11.50  inches,  nearly.      A  comparison 
of  the  first  three  experiments  will   show  that  there   could   be   no    appreciable   differ- 
ence   in   the    results,  that   could   be  attributed  to  the  small  differences  in  the  heights 
of  the   gate,   in   the    experiments   numbered   from    4    to    50,   inclusive,  and   they  are 
accordingly    all    classed    together,   as    experiments    with   a   full    gate,    the    small    dif- 
ference   in    the    heights   being  accidental. 

The  experiments  numbered  from  51  to  64,  inclusive,  were  made  with  the 
gate  raised  8.55  inches,  or  three-fourths  of  the  full  height,  nearly.  Those  num- 
bered from  65  to  76,  inclusive,  were  made  with  the  gate  at  very  nearly  half  of 
the  full  height.  Those  numbered  from  77  to  79,  inclusive,  were  made  with  the 
gate  at  about  seven  eighths  of  the  full  height.  Those  numbered  from  80  to  89, 
inclusive,  were  made  with  the  gate  at  about  one  fourth  of  its  full  height.  And 
the  last  three  experiments  were  made  with  the  gate  raised  one  inch. 

56.  COLUMN  5.     Time.     The  times  entered   under  the  heads  beginning,  and  ending. 
of  the   experiments,  are   taken    from    the    notes   of    the   "speed     of    the    wheel,"    and 
indicate    the    times   at   which    the    bell,  attached    to    the    counter,  was   struck,  which, 
by    a    comparison    of    the    various    note-books,    appeared,    by    the    regularity    of    the 
observations,  to  be    the   most  suitable  for  the  commencement  and  termination  of  the 
experiment. 

4 


26  EXPERIMENTS   UPON  THE   TREMONT  TURBINE. 

57.  COLUMN  6.     Duration  of  the  experiment,  is  obtained   by  taking    the  differences 
of  the  times  of  the"  beginning,  and  ending  of  the  experiment,  as  given  in  column  5. 

58.  COLUMN  7.     Total  number  of  revolutions  of  the  wheel  during  the  experiment.      This 
is   obtained   from   the   note-book    of    the    "  speed    of    the   wheel,"    by   counting    the 
number   of  observations   of  the   times   at  which   the   bell   was  struck;    this   number, 
less   one,   multiplied    by    50,   which   is   the   number   of  revolutions   of  the   wheel   to 
each  stroke  of  the  bell,  gives  the  number  of  revolutions  during  the  experiment. 

59.  COLUMN    8.      Number   of    revolutions   of  the   wheel   per   second,   is   obtained    by 
dividing  the    total    number   of  revolutions    of  the    wheel,   by  the    duration    of    the 
experiment. 

60.  COLUMN  9.     The  weight  in  the  scale,  requires  no  explanation. 

61.  COLUMN  10.      Useful  effect,   or  the  friction   of  the   brake,   in  pounds   avoirdupois, 
raised  one  foot  per  second.     This  is   obtained   by   multiplying  together   the   weight   in 
the   scale,   and   the  velocity   that   the   point   of  application   of  the   weight,   tends   to 
take.      Or,  in   other   words,   the    product   of   the   weight   into    the   velocity   that   the 
weight  would   actually   take,   if,   for    an    infinitely   short    time,   the    brake,   and   the 
apparatus   connecting   it   with   the   weight,  were    rigidly  connected   with   the  friction 
pulley. 

The  effective  length  of  the  brake,  including  the  leverage  due  to  the  different 
lengths  of  the  arms  of  the  bell  crank,  was  10.827778  feet  (art.  50).  The  cir- 
cumference of  a  circle  of  this  radius  is  68.0329  feet.  This  circumference  is  a 
constant  for  all  the  experiments  in  which  any  useful  effect  was  produced,  and 
column  10  was  obtained  by  the  product  of  this  constant,  the  weight,  and  the 
number  of  revolutions  of  the  wheel  per  second.  The  computation  was  performed 
by  logarithms,  and  if  the  results  given  in  the  tables  should  be  verified  by  actual 
multiplications,  minute  differences  Avould,  no  doubt,  be  detected. 

62.  COLUMNS  11  and  12.     Heights   of  the   water  in   the  forebay  and  in  the  wheelptt. 
These  heights  are  all  referred  to  the  top  of  the  weir,  consequently,  the  differences 
give  the  fall  acting  upon  the  wheel. 

63.  COLUMN  13.      Total  fall   acting   upon    the    tvheel.      These    are    the    differences 
referred   to   in   the   last    sentence.      In    experiments   27    and    28,    observations    were 
taken   in  the  ventilating  pipe   G,  plate  I.,  for  the  purpose  of  estimating  the  loss  of 
fall   to  this  part  of  the  supply  pipe ;  —  it  was  not  convenient,  however,  to  measure 
these   heights  with  complete  accuracy.      In  experiment  27,  the   height  of  the  water 
in  the  ventilating  pipe  was  0.106  feet  below  the  level  in  the  forebay ;  —  in  experi- 
ment  28   the    difference   was    found    to    be    0.102    feet;  —  in    experiment    30,   which 
gave    the   maximum   coefficient   of  effect,  the    quantity    of  water   discharged    by   the 
wheel,  was   a   little   less   than   in   either   experiment  27,  or   28.     We  may,  therefore. 


EXPERIMENTS   UPON   THE   TREMONT  TURBINE.  27 

conclude,  that  when  the  regulating  gate  was  fully  raised,  and  the  wheel  running 
with  the  velocity  giving  the  maximum  coefficient  of  effect,  the  fall  acting  .upon 
the  wheel  being  12.903  feet,  the  loss  of  fall  from  the  forebay  to  the  ventilating 
pipe,  was  very  nearly  0.10  feet. 

64.  COLUMN  14.     Depth   of  water   on   the   weir.      The   depths   on   the   weir   were 
observed  with  the  hook  gauge,  described  at  art.  45. 

65.  COLUMN  15.      Quantity  of  water  passing  the  weir.     These    quantities   have   been 
calculated  by  the  formula 

Q  =  33$(l—  O.lnh)$, 
in  which 

Q  =  Quantity,  in  cubic  feet  per  second. 
/  =  The  total  length  of  the  weir,  in  feet. 
n  =  The  number  of  end  contractions  in  the  weir. 
h  =  The  depth  on  the  weir,  in  feet. 

It  is  unnecessary  here  to  discuss  the  reasons  that  have  induced  the  author  to 
adopt  this  formula,  sa  different  from  any  that  has  been  used  heretofore,  as  the 
subject  is  fully  considered  in  another  part  of  this  work. 

A  small  quantity  of  water  entered  the  wheelpit  without  passing  through  the 
wheel ;  there  was  also  a  small  quantity  that  leaked  out  by  passing  through  the  floor 
of  the  wheelpit;  the  latter  quantity,  when  the  depth  on  the  weir  was  0.496  feet, 
was  estimated  at  0.0409  cubic  feet  per  second;  see  art.  130.  As  these  quantities 
were  very  minute,  and  tended  to  compensate  each  other,  they  have  been  neglected, 
and  the  quantity  computed  as  passing  the  weir  is  taken  for  the  quantity  discharged 
by  the  wheel. 

66.  COLUMN  16.     Total  power  of  the  ivater.      This   column   is   obtained    by   multi- 
plying  together   the   total   fall  acting  upon  the  wheel,  the  quantity  of  water  passing 
the   weir  per  second,  and  the  weight   of  a   cubic  foot  of  water.      The  temperature 
of  the   water  was  constantly  at  32°  Fahrenheit,  it  was  nearly  pure,  and  the  weight 
of  a  cubic  foot  was  taken  at  62.375  pounds  avoirdupois. 

The  water  of  the  Merrimack  Eiver  is  always  remarkably  free  from  impurities, 
held  in  solution,  flowing,  as  it  does,  from,  and  through  a  primitive  formation,  cov- 
ered with  a  sterile  soil.  In  midwinter,  at  which  season  these  experiments  were 
made,  it  is  more  than  ordinarily  pure,  as  at  that  season  the  surface  of  the  coun- 
try is  usually  covered  with  snow,  and  the  soil  frozen  to  a  considerable  depth ; 
the  river  itself,  wherever  it  flows  with  a  moderate  current,  is  frozen  over,  so  that 
heavy  carriages  can  often  pass  with  safety,  and  at  the  time  when  these  experi- 
ments were  made,  the  river  for  about  eighteen  miles  before  it  reached  the  turbine, 


28  EXPERIMENTS   UPON   THE   TREMONT   TURBINE. 

was  covered  with  a  solid  coating  of  ice,  with  scarcely  an  opening  in  the  whole 
distance.  When  the  river  is  thus  frozen,  the  water  flows  along  under  the  ice, 
entirely  free  from  floating  particles  of  ice,  even  in  the  most  severe  weather. 

As  the  author  had  frequently  felt  the  want  of  a  table  of  the  absolute  weights 
of  a  cubic  foot  of  water  at  different  temperatures,  he,  several  years  since,  com- 
puted the  following  table. 

In  the  Encyclopedia  Britannica,  seventh  edition,  vol.  21,  page  846,  is  given  the 
following  extract  from  the  British  act  of  Parliament,  establishing  the  standards  fpr 
weights  and  measures. 

"Provided  always,  and  be  it  enacted,  that  in  all  cases  of  dispute  respecting  the 
correctness  of  any  measure  of  capacity,  arising  in  a  place  where  recourse  cannot 
conveniently  be  had  to  any  of  the  aforesaid  verified  copies  or  models  of  the 
standard  measures  of  capacity,  it  shall  and  may  be  lawful,  to  and  for,  any  justice 
of  the  peace,  or  magistrate,  having  jurisdiction  in  such  place,  to  ascertain  the  con- 
tent of  such  measure  of  capacity  by  direct  reference  to  the  weight  of  pure  or 
rain  water  which  such  measure  is  capable  of  containing;  ten  pounds  avoirdupois 
weight  of  such  water,  at  the  temperature  of  62°  by  Fahrenheit's  thermometer,  being 
the  standard  gallon  ascertained  by  this  act,  the  same  being  in  bulk  equal  to 
277.276,  1822  (1823,  277.274)  -cubic  inches,  and  so  in  proportion,"  etc.  277.274 
cubic  inches  was  taken,  as  it  appeared  to  be  the  latest  determination. 

In  the  first  volume  of  the  Traite  de  Chinue,  by  J.  J.  Berzetius,  second  French 
edition,  Paris,  1846,  there  is  given  a  table  of  the  specific  gravities  of  pure  water,  at 
different  temperatures  of  the  centigrade  scale,  deduced  from  Haellstroem's  experi- 
ments. 

From  these  two  authorities  were  derived  the  data  for  the  following  table. 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 


29 


TABLE   I. 


WEIGHT  OF  A  CUBIC  FOOT  OF  PURE  WATER  AT  DIFFERENT   TEMPERATURES. 


Temperature, 
in  degrees  of 
Fahrenheit's 
thermometer. 

Weight  in  air,  of  a 
cubic  foot  of  pure 
water.     Founds 
avoirdupois. 

Temperature, 
in  degrees  of 
Fahrenheit's 
thermometer. 

Weight  hi  air,  of  a 
cubic  foot  of  pure 
water.    Pounds 
avoirdupois. 

Temperature, 
in  degrees  of 
Fahrenheit's 
thermometer. 

Weight  in  air,  of  a 
cubic  foot  of  pure 
water.     Pounds 
avoirdupois. 

32 

62.375 

50 

62.368 

69 

62.278 

33 

62.377 

51 

62.365 

70 

62.272 

34 

62.378 

52 

62.363 

71 

62.264 

35 

62.379 

53 

62.359 

72 

62.257 

36 

62.380 

54 

62.356 

73 

62.249 

37 

62.381 

55 

62.352 

74 

62.242 

38 

62.381 

56 

62.349 

75 

62.234 

39  ,„.., 

62.382 

57 

62.345 

76 

62.225 

39.38 

62.382 

58 

62.340 

77 

62.217 

40 

62.382 

59 

62.336 

78 

62.208 

41 

62.381 

60 

62.331 

79 

62.199 

42 

62.381 

61 

62.326 

80 

62.190 

43 

62.380 

62 

62.321 

81 

62.181 

44 

62.379 

63 

62.316 

82 

62.172 

45 

62.378 

64 

62.310 

83 

62.162 

46 

62.376 

65 

62.304 

84 

62.152 

47 

62.375 

66 

62.298 

85 

62.142 

48 

62.373 

67 

62.292 

86 

62.132 

49 

62.371 

68 

62.285 

67.  COLUMN  17.      Ratio  of  the  useful  effect  to  the  power  expended.      This  column  is 
obtained  by  dividing  the  numbers  in  column  10  by  those  in  column  16. 

68.  COLUMN  18.     Velocity  due  to  the  fall  acting  upon  the  wheel.      The   numbers   in 
this  column  have  been  calculated  by  the  formula 


V=  the  velocity  in  feet  per  second. 

^  =  the  velocity  acquired  by  a  body  at  the  end  of  the  first  second   of  its  fall 

in  a  vacuum. 
h  =  the  fall  acting  upon  the  wheel  j    this  is  given  in  column  13. 

The   value   of  g  has    been    calculated    by  the    formula    given    in    the    second 
edition  of  the  Trait6  D'HydrauKque,  by  D'Aubuisson,  page  5,  viz.  :  — 

g  =  9!"  8051  (1  —  0.00284  cos.  2  f)  (1  —  if  )  ; 

/being  the  latitude  of  the-  place;    e,  its  elevation  above  the  level  of  the  sea; 
and  r,  the  radius  of  the  terrestrial  spheroid,  at  the  level  of  the  sea,  and  at  the  place  ; 

{  r=  6366407"1  (1  +  0.00164  cos.  21.)}. 


30  EXPEEIMENTS  UPON  THE  TREMONT  TURBINE. 

The  latitude  of  Lowell,  as  given  in  the  American  Almanac,  is  42°,  38',  46", 
and  the  height  above  the  sea  is  known  to  be  about  25  metres.  With  these  data, 
the  above  formula  gives,  in  feet, 

^  =  32.1618. 

69.  COLUMN  19.      Velocity   of  the  inferior   circumference   of  the  wheel.     The  diameter 
of  the  circle   inscribing   the   inner   edges   of  the   buckets,  is   6.75  feet;    see   art.   35. 
Consequently  the   interior  circumference  of  the  wheel  is  21.20575  feet.     The  product 
of    this   number   into   the    number    of    revolutions   per   second,   given   in   column   8, 
gives  the  numbers  in  column  19. 

70.  COLUMN  20.     Ratio   of  the   velocity  of  the  interior  circumference   of  the  wheel,  to 
the  velocity  due  to  the  fall  acting  on   the  wheel.      This    column   is   obtained   by   dividing 
the   numbers   in   column    19    by   the    corresponding    numbers    in   column    18.      This 
column   indicates   the   relative   velocities   of  the  wheel,  in   the   different  experiments, 
eliminated    from   the    effects   of  the   variations   in   the   fall   acting   upon  the  wheel. 

71.  COLUMN  21.      Quantity   of  ivater   which    passed  the   wheel,   reduced  to   a   uniform 
fall  of  thirteen  feet.     The   numbers   in   this   column   are    obtained   from   those   in  col- 
umn 15,  in  the  following  manner. 

Let  IT=  the  observed  fall  acting  upon  the  wheel. 
Q  =  the  observed  quantity. 

(X=the    quantity  that  would   have   passed  the  wheel,  if  the  fall   had   been 
thirteen  feet,  instead  of  H,  all  other  circumstances  being  the  same. 

As  the  quantity  of  water  discharged  by  the  wheel,  all  other  things  being 
equal,  will  vary  as  the  square  root  of  the  fall  acting  upon  the  wheel,  we  have 

V/JT:    Q    =:    ViT  :     <?, 
therefore  Qf  =  Q   /". 


The   quantities  given  in   column   21,  have   been   calculated   by  this  formula. 

72.  COLUMN  22.  Ratio  of  the  reduced  quantity  in  column  21,  to  the  reduced  quantity 
in  experiment  30.  The  numbers  given  in  this  column  indicate  the  relative  quantities 
discharged  by  the  wheel  in  the  different  experiments,  eliminated  from  the  effects 
due  to  the  variations  in  the  fall  acting  upon  the  wheel  ;  the  reduced  quantity 
in  experiment  30  is  taken  as  unity,  that  experiment  giving  the  maximum  coeffi- 
cient of  effect.  It  will  be  seen  by  a  comparison  of  columns  20  and  22,  that  the 
quantity  discharged  by  the  wheel,  when  the  gate  is  fully  raised,  diminishes  regu- 
larly with  the  velocity.  The  quantity  discharged  is  a  minimum  in  experiment  42, 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE.  31 

m  which  the  wheel  had  the  least  velocity.  In  experiments  43  and  44,  however, 
in  which  the  wheel  was  prevented  from  revolving,  by  screwing  up  the  brake,  the 
quantity  discharged  was  considerably  above  the  minimum.  Whether  this  is  due 
to  an  accidental  position  of  the  buckets  relative  to  the  guides,  presenting  aper- 
tures more  favorable  to  the  discharge  than  the  average  of  all  positions,  or  whether 
it  is  due  to  some  more  general  cause,  the  author  is  not  aware. 

73.  COLUMN  23.     Direction   of  the   water   having  the  tvheel,  as   indicated  by  the  vane. 
The   angles   given   in   this  column  show  the  position   of  the  vane,  relative  to  a  line 
passing   through   the   axis    of    the   vane,    and   parallel    to   a   tangent   drawn   through 
the   point  in    the    circumference    of   the   wheel,   nearest  to    the   axis   of  the   vane, 
zero   being  in   the    direction    of    the    motion    of   the   wheel.      The   apparatus   with 
which  these  angles  were  taken,  is  described  at  art.  43.      In  the   experiments  made 
when   the  gate  was  fully  raised,  or  nearly  so,-  the  vane  operated  satisfactorily ;    but 
as  the   height    of    the    gate   was   diminished,   the   indications    of    the   vane    became 
more    uncertain.      The   vane   was   of  nearly   the    same  height  as   the  orifices  in  the 
exterior   circumference   of  the   wheel;    this   was   very   suitable   for    the    experiments 
with   the   gate   fully  raised,  but  in    the    experiments  with  the   gate   partially  raised, 
a   portion   of    the    height    of    the   vane   was    exposed    to    irregular    currents,   which 
probably  interfered   seriously   with   its   operation.      The   observations   made  with   the 
vane   in   the   experiments  in  which    the    gate    was    partially   raised,   are   much   less 
to  be   relied    on  than   those    made   when  the   gate   was  fully  raised,   the   value   of 
the   indications  being,  in  some  degree,  proportioned  to  the  height  of  the  gate. 

74.  COLUMN  24.      Mean   elevation   of  the  pointer   on   the   bell  crank.      The    numbers 
in   this   column   indicate   the   mean   positions   of  the   bell   crank,  during  the   experi- 
ments,  in   reference   to   a   gauge    placed    6.5    feet    from    the    fulcrum    of  the    bell 
crank.      It  will    be    seen    by   the   table   that    the   mean   positions    differ   but   little 
from  the   horizontal ;    the    pointer    was    however    generally   a    little    below,   which 
indicates  that  the   weight   was   generally   lifted   a   little   too   high. 

The  play  of  the  brake  was  confined  between  two  fixed  stops,  placed  so  that 
when  the  pointer  stood  at  0.20  feet  below  the  horizontal,  the  brake  struck, — 
and  it  struck  the  other  stop  when  the  pointer  was  at  0.21  feet  above  the  hori- 
zontal. The  brake  was  not  allowed,  however,  to  touch  either  of  the  stops  in  any 
of  the  experiments  reported,  in  which  it  was  undertaken  to  regulate  the  friction 
of  the  brake ;  the  fact  that  it  did  touch  was  deemed  a  sufficient  reason  to  reject 
the  experiment.  Little  inconvenience,  however,  was  experienced  from  this  cause, 
as  the  hydraulic  regulator  afforded  very  perfect  control  over  the  brake. 


32 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 


TABLE 

EXPERIMENTS  UPON  THE  TURBINE  AT  THE 


1 

3 

3 

4 

5 

6 

7 

8 

9 

1O 

Temperature  of 

TIME. 

Total 

the  atmosphere  in 

number 

Useful  effect. 

No. 

' 

degrees  of      • 
Fahrenheit's 

Height 

Duration 

of 

Number  of 

Weight  In 

or  the 

of  the 

DATE 

thermometer. 

of  the 

of  the 

tions 

revolutions 

the  scale, 

the  brake, 

experi- 
ment. 

1851. 

External 
air  In 
the 

In  the 

wheelpit. 

regulat- 
ing gate, 
In  Inches. 

Beginning  of  the 
experiment. 

d  Ending  of  the 

experiment. 

experi- 
ment, 
In  seconds. 

of  the 
wheel 
during 
the 

of  the  wheel 
per  second. 

in  pounds 
avoirdupois. 

in  pounds 
avoirdupois, 
raised  one 
foot  per 

shade. 

H. 

mln. 

sec. 

H. 

IP  in. 

sec. 

ment. 

1 

February  17,  P.M. 

31.00 

41.00 

11.50 

2 

19 

52.00 

2 

28 

15.50 

503.50 

450 

0.89374 

1443.34 

87760.8 

2 

it             a        a 

29.00 

36.75 

11.60 

2 

37 

8.00 

2 

47 

24.75 

616.75 

550 

0.89177 

1443.34 

87567.2 

3 

u             u        u 

30.25 

36.25 

11.45 

2 

56 

26.00 

3 

6 

41.50 

615.50 

650 

0.89358 

1443.34 

87745.0 

4 

a            u        u 

29.00 

35.25 

11.49 

3 

23 

24.00 

3 

29 

8.50 

344.50 

550 

1.59651 

307.03 

33348.3 

5 

u            u        u 

29.50 

35.50 

ft 

3 

29 

8.50 

3 

37 

18.00 

489.50 

750 

1.53218 

411.48 

42892.0 

6 

u            u        u 

29.75 

35.25 

U 

3 

37 

18.00 

3 

44 

42.75 

444.75 

650 

1.46149 

519.77 

51680.6 

7 

a             u        u 

29.50 

35.50 

ft 

3 

45 

18.00 

3 

52 

32.00 

434.00 

600 

1.38249 

638.36 

60040.8 

8 

u             a        u 

29.25 

35.50 

U 

4 

4 

35.00 

4 

9 

40.50 

305.50 

400 

1.30933 

750.42 

66845.5 

9 

u             u        u 

29.25 

35.50 

it 

4 

10 

19.75 

4 

15 

41.00 

321.25 

400 

1.24514 

854.87 

72416.3 

10 

u             a        tt 

29.00 

35.50 

u 

4 

15 

41.00 

4 

24 

7.50 

506.50 

600 

1.18460 

957.35 

77154.6 

11 

u             u        a 

29.00 

35.75 

H 

4 

24 

51.00 

4 

33 

44.25 

533.25 

600 

1.12518 

1057.49 

80949.8 

12 

ft             t(        ti 

28.50 

35.00 

u 

5 

1 

10.50 

5 

10 

31.00 

560.50 

1000 

1.78412 

0. 

0. 

13 

February  18,  A.M. 

35.75 

16 

9 

14 

5.50 

9 

22 

58.00 

532.50 

950 

1.78404 

0. 

0. 

14 

u             u        u 

33.75 

36.25 

It 

9 

42 

32.00 

9 

51 

7.25 

515.25 

550 

1.06744 

1156.27 

83969.8 

15 

tt              ft         ft 

34.25 

36.75 

U 

9 

51 

7.25 

10 

0 

4.50 

537.25 

550 

1.02373 

1229.41 

85625.3 

16 

u             u        u 

34.00 

36.50 

tt 

10 

12 

27.00 

10 

19 

57.25 

450.25 

450 

0.99945 

1269.42 

86314.4 

17 

U                  U           It 

33.75 

36.50 

0 

10 

20 

48.00 

10 

29 

23.50 

515.50 

500 

0.96993 

1319.22 

87051.8 

18 

u             tt        u 

34.00 

36.50 

u 

10 

41 

55.00 

10 

48 

58.25 

423.25 

400 

0.94507 

1359.23 

87392.7 

19 

tt              tt         tt 

34.75 

36.75 

tt 

10 

49 

52.00 

10 

59 

48.00 

596.00 

550 

0.92282 

1397.12 

87714.1 

20 

u            u        <t 

36.00 

36,00 

tt 

11 

16 

14.50 

11 

25 

23.25 

548.75 

500 

0.91116 

1416.70 

87819.8 

21 

tl            tt        tt 

36.50 

36.50 

tl 

11 

25 

23.25 

11 

35 

33.00 

609.75 

550 

0.90201 

1433.43 

87964.3 

22 

tt              tt         If 

36.50 

36.75 

tt 

11 

45 

12.00 

11 

59 

8.00 

836.00 

750 

0.89713 

1443.06 

88076.2 

23 

February  18,  P.M. 

41.50 

39.25 

tt 

2 

23 

56.50 

2 

33 

18.00 

561.50 

1000 

1.78094 

0. 

0. 

24 

tt          tt      tt 

39.75 

38.75 

tt 

2 

41 

30.50 

2 

50 

51.00 

560.50 

500 

0.89206 

1454.24 

88257.1 

25 

u             u        u 

39.00 

40.00 

tt 

2 

50 

51.00 

3 

4 

2.00 

791.00 

700 

0.88496 

1464.80 

88189.9 

26 

tt              tt         tt 

38.75 

38.00 

it 

3 

22 

7.50 

3 

26 

53.25 

285.75 

250 

0.87489 

1474.37 

87756.6 

27 

tt              tt         tt 

38.75 

38.00 

it 

3 

27 

54.00 

3 

42 

6.25 

852.25 

750 

0.88002 

1474.37 

88271.4 

28 

It                  U           U 

38.50 

36.25 

it 

3 

58 

40.25 

4 

11 

4.75 

744.50 

650 

0.87307 

1485.63 

88242.7 

29 

tl                  tl           U 

38.50 

36.25 

it 

4 

28 

54.50 

4 

40 

27.00 

692.50 

600 

0.86643 

1498.66 

88339.3 

30 

tt              tt         tt 

37.25 

36.50 

tt 

4 

55 

58.00 

5 

7 

43.00 

705.00 

600 

0.85106 

1524.67 

88278.9 

31 

February  20,  A.M. 

11.48 

9 

16 

16.25 

9 

25 

35.00 

558.75 

1000 

1.78971 

0. 

0. 

32 

u          tt       u 

33.50 

35.00 

tt 

9 

47 

40.50 

9 

53 

39.25 

358.75 

300 

0.83624 

1552.44 

88320.8 

33 

u          tt      tt 

33.75 

35.00 

tt 

10 

8 

35.00 

10 

18 

49.75 

614.75 

500 

0.81334 

1597.08 

88372.5 

34 

U                  It           It 

36.75 

35.50 

tt 

10 

37 

30.50 

10 

48 

8.25 

637.75 

500 

0.78401 

1648.87 

87947.8 

35 

U                  It           U 

38.00 

35.75 

tt 

11 

2 

8.50 

11 

13 

22.25 

673.75 

500 

0.74211 

1724.49 

87066.5 

36 

ft              tt         tt 

41.00 

35.75 

tt 

11 

23 

38.25 

11 

34 

26.00 

647.75 

450 

0.69471 

1816.71 

85863.7 

37 

U                  tl           U 

41.50 

36.00 

u 

11 

48 

56.00 

11 

59 

15.50 

619.50 

400 

0.64568 

1911.45 

83965.5 

38 

February  20,  P.M. 

ti 

2 

30 

39.50 

2 

39 

59.50 

560.00 

1000 

1.78571 

0. 

0. 

39 

it          tt       tt 

42.25 

35.25 

it 

2 

55 

11.00 

3 

6 

46.50 

695.50 

450 

0.64702 

1911.45 

84139.1 

40 

U                  U           tl 

41.75 

35.75 

it 

3 

21 

56.50 

3 

30 

16.50 

500.00 

300 

0.60000 

2011.52 

82109.8 

41 

It              ft         It 

41.75 

35.75 

tt 

3 

45 

27.50 

3 

56 

25.00 

657.50 

350 

0.53232 

2167.38 

78492.2 

42 

tt              It         tt 

41.50 

36.00 

it 

4 

9 

29.00 

4 

18 

39.50 

550.50 

250 

0.45413 

2367.88 

73158.0 

43 

It              It         tt 

it 

4 

58 

0. 

4 

59 

30.00 

90.00 

0 

0. 

4213.38 

0. 

44 

ft              It         tl 

tt 

5 

2 

0. 

5 

4 

30.00 

150.00 

0 

0. 

3946.38 

0. 

45 

February  21,  A.M. 

36.25 

35.50 

it 

9 

22 

57.50 

9 

32 

18.25 

560.75 

1000 

1.78333 

0. 

0. 

46 

u          'tt       tt 

36.25 

35.75 

u 

9 

38 

3.00 

9 

49 

4.00 

661.00 

550 

0.83207 

1565.21 

88603.9 

47 

n           u      ti 

36.25 

35.75 

it 

10 

0 

48.75 

10 

11 

1.00 

612.25 

500 

0.81666 

1590.50 

88367.8 

48 

u          tt       tt 

35.75 

36.25 

u 

10 

25 

38.50 

10 

37 

3.25 

684.75 

550 

0.80321 

1614.79 

88240.1 

49 

ti          u       tt 

36.00 

36.00 

n 

10 

50 

35.00 

11 

2 

11.75 

696.75 

550 

0.78938 

1641.34 

88146.2 

50 

It                  U           U 

35.75 

36.25 

n 

11 

14 

54.00 

11 

25 

44.25 

650.25 

500 

0.76893 

1679.62 

878*5.8 

EXPERIMENTS   UPON  THE   TREMONT  TURBINE. 


33 


IF. 

TRKMONT  MILLS,  IN  LOWELL,   MASSACHUSETTS. 


11 

13 

13         14 

15 

1G 

17 

18 

19 

3O 

31 

33 

33 

34 

Height 
of  the 

Height 
of  the 

Total 

Quantity 

Total  power 

Ratio 

Velocity 

Telocity 

Ratio 
of  the 

Quantity  of 
water  which 

Ratio 
of  the 

Direction 

Mean 

No. 

water 

water 
after 

fall 

Depth  of 

of  water 

of  the  water, 

of  the 

due  to  the 

of  the 

velocity  of 
the  interior 

passed  the 

reduced 

of  the 
water 

elevation 

of  the 

above  the 

passing 

water 

in  pounds 

fall  acting 

interior 

circumfnce 

wheel, 

quantity  in 

leaving 

of  the 

experi- 

wheel, 
taken  in 

the 
wheel, 
taken 

acting 
upon  the 

on  the 

the/  weir,  in 

avoirdupois 

useful 
effect  to 

on  the 
wheel  ill 

circumf  nee 
of  the 

of  the 
wheel  to 
the 

reduced  to  a 
uniform  fall 

column  21, 
to  the 

the  wheel, 
as 
indicated 

pointer  on 

ment. 

the 

in  the 

wheel, 

weir, 

cubic  feet 

raised  one 

the  power 

wheel,  in 

velocity 

of  13  feet,  in 

reduced 

by  the 

the  bell 

forebay, 

wheel- 
nit 

in  feet. 

in  feet. 

per  second. 

foot  per 

expended. 

feet  per 

feet  per 

due  to  the 

cuhic  feet 

quantity  in 

vane. 

crank. 

f    f 

pit, 

second. 

second. 

second. 

^^     ii 

per  second. 

experiment 

in  feet. 

on  the 
wheel. 

30. 

deg. 

m. 

Feet. 

i 

15.082 

2.218 

12.864 

1.8811 

139.4206 

111870.0 

0.78449 

28.7656 

18.9525 

0.65886 

140.1557 

1.01044 

47 

(i 

+0.002 

2 

15.079 

2.219 

12.860 

1.8811 

139.4206 

111835.2 

0.78300 

28.7611 

18.9107 

0.65751 

140.1775 

1.01060 

47 

•15 

0. 

3 

15.087 

2.218 

12.869 

1.8816 

139.4676 

111951.2 

0.78378 

28.7712 

18.9491 

0.65861 

140.1756 

1.01058 

47 

23 

—0.001 

4 

15.036 

2.482 

12.554 

2.0383 

156.6470 

122663.3 

0.27187 

28.4169 

33.8553 

1.19138 

159.4053 

1.14922 

+0.013 

5 

15.042 

2.431 

12.611 

2.0180 

154.3891 

121444.2 

0.35318 

28.4813 

32.4909 

1.14078 

156.7522 

1.13009 

-f-0.008 

6 

15.051 

2.398 

12.653 

1.9989 

152.2682 

120174.8 

0.43005 

28.5287 

30.9921 

1.08635 

154.3421 

1.11271 

—0.003 

7 

15.061 

2.365 

12.696 

1.9734 

149.4653 

118363.5 

0.50726 

28.5771 

29.3167 

1.02588 

151.2442 

1.09038 

15 

•1'.) 

—0.001 

8 

15.069 

2.349 

12.720 

1.9536 

147.2942 

116864.8 

0.57199 

28.6041 

27.7653 

0.97067 

148.9066 

1.07353 

18 

8 

—0.010 

9 

15.089 

2.312 

12.777 

1.9420 

146.0204 

116373.2 

0.62228 

28.6681 

26.4041 

0.92102 

147.2891 

1.06187 

20 

8 

—0.018 

10 

15.102 

2.302 

12.800 

1.9315 

144.8734 

115667.0 

0.66704 

28.6939 

25.1203 

0.87546 

146.0009 

1.05258 

22 

66 

—0.013 

11 

15.100 

2.281 

12.819 

1.9226 

143.9088 

115067.3 

0.70350 

28.7152 

23.8602 

0.83093 

144.9212 

1.04480 

25 

17 

—0.001 

12 

15.028 

0. 

37.8336 

13 

15.071 

2.561 

12.510 

2.0989 

163.4313 

127527.3 

0. 

28.3670 

37.8319 

1.33366 

166.6013 

1.20110 

14 

15.120 

2.264 

12.856 

1.9098 

142.5180 

114284.2 

0.73475 

28.7566 

22.6359 

0.78716 

143.3140 

1.03321 

29 

49 

+0.002 

15 

15.117 

2.229 

12.888 

1.9054 

142.0433 

114187.1 

0.74987 

28.7924 

21.7090 

0.75398 

142.6592 

1.02849 

33 

30 

+0.004 

16 

15.116 

2.226 

12.890 

1.9048 

141.9762 

114150.8 

0.75614 

28.7946  21.1940 

0.73604 

142.5808 

1.02792 

35 

87 

—0.001 

17 

15.128 

2.232 

12.896 

1.8983 

141.2762 

113640.9 

0.76603 

28.8013 

20.5681 

0.71414 

141.8448 

1.02262 

38 

•2( 

—0.001 

18 

15.111 

2.231 

12.880 

1.8908 

140.4657 

112848.8 

0.77442 

28.7835 

20.0409 

0.69626 

141.1186 

1.01738 

41 

26 

—0.003 

19 

15.114 

2.231 

12.883 

1.8873 

140.0845 

112568.7 

0.77920 

28.7868 

19.5691 

0.67979 

140.7192 

1.01450 

44 

21 

4-0.001 

20 

15.114 

2.228 

12.886 

1.8866 

140.0066 

112532.3 

0.78040 

28.7902 

19.3219 

0.67113 

140.6246 

1.01382 

46 

18 

—0.002 

21 

15.128 

2.229 

12.899 

1.8856 

139.9040 

112563.3 

0.78147 

28.8047 

19.1278 

0.66405 

140.4507 

1.01257 

47 

26 

—0.005 

22 

15.123 

2.225 

12.898 

1.8834 

139.6678 

112364.5 

0.78384 

28.8036 

19.0243 

0.66048 

140.2190 

1.01090 

48 

26 

0. 

23 

14.965 

2.536 

12.429 

2.0834 

161.6944 

125355.0 

0. 

28.2750 

37.7662 

1.33567 

165.3669 

1.19220 

24 

15.124 

2.219 

12.905 

1.8773 

139.0070 

111893.5 

0.78876 

28.8114 

18.9168 

0.65657 

139.5177 

1.00584 

49 

L>S 

—0.006 

25 

15.118 

2.219 

12.899 

1.8775 

139.0291 

111859.4 

0.78840 

28.8047 

18.7662 

0.65150 

139.5724 

1.00623 

50 

87 

40.004 

26 

15.102 

2.209 

12.893 

1.8750 

138.7601 

111591.0 

0.78641 

28.7980 

18.5527 

0.64424 

139.3347 

1.00452 

51 

•  lo 

—0.057 

27 

15.116 

2.214 

12.902 

1.8758 

138.8489 

111740.3 

0.78997 

28.8080 

18.6616 

0.64779 

139.3752 

1.00481 

51 

18 

—0.018 

28 

15.117 

2.211 

12.906 

1.8760 

138.8711 

111792.9 

0.78934 

28.8125 

18.5141 

0.64257 

139.3759 

1.00482 

52 

22 

—0.018 

29 

15.118 

2.212 

12.906 

1.8727 

138.5134 

111504.9 

0.79225 

28.8125 

18.3732 

0.63768 

139.0169 

1.00223 

53 

52 

—0.017 

30 

15.111 

2.208 

12.903 

1.8697 

138.1892 

111218.1 

0.79375 

28.8092 

18.0474 

0.62645 

138.7076 

1.00000 

58 

10 

—0.018 

31 

15.084 

2.545 

12.539 

2.0891 

162.3283 

126960.2 

0. 

28.3999 

37.9521 

1.33635 

165.2853 

1.19161 

32 

15.119 

2.204 

12.915 

1.8704 

138.2668 

111384.0 

0.79294 

28.8225 

17.7330 

0.61525 

138.7211 

1.00010 

61 

54 

—0.011 

33 

15.134 

2.200 

12.934 

1.8701 

138.2335 

111521.1 

0.79243 

28.8437 

17.2475 

0.59796 

138.5858 

0.99912 

66 

5 

—0.008 

34 

15.129 

2.188 

12.941 

1.8687 

138.0869 

111463.0 

0.78903 

28.8515 

16.6254 

0.57624 

138.4013 

0.99779 

86 

12 

—0.004 

35 

15.123 

2.184 

12.939 

1.8652 

137.7076 

111139.7 

0.78340 

28.8493 

15.7371 

0.54549 

138.0318 

0.99513 

99 

25 

—0.014 

30 

15.128 

2.184 

12.944 

1.8539 

136.4917 

110200.9 

0.77916 

28.8549 

14.7319 

0.51055 

136.7866 

0.98615J115 

is 

—0.001 

37 

15.117 

2.177 

12.940 

1.8412 

135.1415 

109077.1 

0.76978 

28.8504 

13.6922 

0.47459 

135.4545 

0.97655 

131 

18 

—0.013 

38 

15.036 

2.536 

12.500 

2.0834 

161.6944 

126071.1 

0. 

28.3557 

37.8674 

1.33544 

164.8966 

1.18881 

39 

15.143 

2.180 

12.963 

1.8431 

135.3423 

109433.4 

0.76886 

28.8761 

13.7205 

0.47515 

135.5354 

0.97713 

130 

51 

—0.019 

40 

15.136 

2.163 

12.973 

1.8380 

134.7976 

109077.0 

0.75277 

28.8872 

12.7235 

0.44045 

134.9378 

0.97282 

139 

45 

—0.010 

41 

15.136 

2.159 

12.977 

1.8282 

133.7538 

108265.8 

0.72499 

28.8916 

11.2882 

0.39071 

133.8723 

0.96514 

147 

25 

—0.005 

42 

15.133 

2.185 

12.948 

1.8252 

133.4330 

107764.7 

0.67887 

28.8593 

9.6302 

0.33370 

133.7007 

0.96390 

—0.015 

43 

15.116 

2.319 

12.797 

1.8460 

135.6536 

108280.4 

0. 

28.6906 

0. 

0. 

136.7253 

0.98571 

44 

15.096 

2.322 

12.774 

1.8457 

135.6205 

108059.4 

0. 

28.6648 

0. 

0. 

136.8150 

0.98635 

45 

15.012 

2.541 

12.471 

2.0864 

162.0237 

126034.8 

0. 

28.3228 

37.8168 

1.33521 

165.4245 

1.19261 

46 

15.156 

2.202 

12.954 

1.8737 

138.6244 

112009.3 

0.79104 

28.8660 

17.6447 

0.61126 

138.8703 

1.  001  17 

60 

52 

—0.012 

47 

15.134 

2.202 

12.932 

1.8726 

138.5023 

111720.6 

0.79097 

28.8415 

17.3179 

0.60045 

138.8660 

1.00114 

63 

16 

—0.008 

48 

15.142 

2.194 

12.948 

1.8723 

138.4690 

111832.0 

0.78904 

28.8593 

17.0327 

0.59020 

138.7468 

1.00028 

66 

27 

—0.01  6 

49 

15.1  44!  2.193i  12.951 

1.8714 

138.3692 

111777.2 

0.78859 

28.8627 

16.7394 

0.57997 

138.6307 

0.99945 

81 

IS 

—0.007 

50    1  5.1  44!  2.1  92J  12.952 

1.8694 

138.1559 

111613.5 

0.78723 

28.8638 

16.3058 

0.56492 

138.4117 

0.99787 

89 

. 

11 

—0.010 

34 


EXPERIMENTS   UPON   THE   TREMONT  TURBINE. 


TABLE 

EXPERIMENTS  UPON  THE  TURBINE  AT  THE 


1 

a 

3 

4 

5 

6 

7 

8 

9 

1O 

Temperature  of 

TIME. 

Total 

the  atmosphere  in 

number 

Useful  effect, 

No. 

degrees  of 

Height 

Duration 

of 

Number  of 

Weight  In 

or  the 

of  the 

DATE, 

Fahrenheit's 
thermometer. 

of  the 

of  the 

revolu- 
tions 

revolutions 

the  scale, 

friction  of 
the  brake, 

experi- 
ment. 

1851. 

External 
air  in 

In  the 

regulat- 
ing gate, 

Beginning  of  the 
experiment. 

Ending  of  the 
experiment. 

experi- 
ment, 

of  the 
wheel 
during 
the 

of  the  wheel 
per  second. 

in  pounds 
avoirdupois. 

in  pounds 
avoirdupois, 
raised  one 
foot  per 

the 

wheelpit. 

second. 

shade. 

ment. 

H. 

mln. 

sec. 

H. 

mln. 

sec. 

51 

February  21,  P.M. 

34.75 

36.50 

8.55 

2 

17 

11.50 

2 

23 

56.00 

404.50 

600 

1.48331 

390.95 

39452.4 

52 

ii              tt         u 

34.50 

36.25 

it 

2 

24 

31.00 

2 

32 

28.00 

477.00 

600 

1.25786 

775.61 

66373.6 

53 

(t              it        a 

34.50 

36.50 

a 

2 

33 

10.00 

2 

41 

55.50 

525.50 

600 

1.14177 

963.30 

74827.2 

54 

t(                   It           U 

34.50 

36.50 

ti 

2 

41 

55.50 

2 

50 

25.00 

509.50 

550 

1.07949 

1069.05 

78512.0 

55 

tf                     it             11 

34.50 

37.00 

u 

2 

50 

25.00 

2 

58 

33.00 

488.00 

500 

1.02459 

1150.77 

80215.4 

56 

11                     it             It 

34.50 

36.75 

a 

3 

8 

34.00 

3 

16 

20.50 

466.50 

450 

0.96463 

1242.98 

81572.6 

57 

tl             tt        n 

34.50 

37.00 

u 

3 

17 

13.50 

3 

25 

17.00 

483.50 

450 

0.93071 

1293.63 

81911.6 

58 

u             11        tt 

34.25 

37.00 

u 

3 

25 

17.00 

3 

33 

37.00 

500.00 

450 

0.90000 

1345.47 

82382.7 

59 

it             u        u 

34.25 

37.25 

u 

3 

33 

37.00 

3 

42 

17.00 

520.00 

450 

0.86538 

1396.11 

82195.5 

60 

u              u         u 

34.25 

36.75 

tt 

3 

43 

15.50 

3 

52 

14.25 

538.75 

450 

0.83527 

1444.03 

82057.9 

61 

u              a        (£ 

34.25 

36.50 

tt 

4 

10 

26.00 

4 

19 

45.50 

559.50 

450 

0.80429 

1494.68 

81786.2 

62 

ti             u        « 

34.00 

36.00 

tl 

4 

31 

37.50 

4 

40 

15.50 

518.00 

400 

0.77220 

1548.69 

81360.6 

63 

u              a         a 

34.00 

36.00 

It 

4 

51 

21.50 

4 

59 

36.00 

494.50 

350 

0.70779 

1656.98 

79788.1 

64 

u               u         a 

tt 

5 

8 

37.00 

5 

19 

17.25 

640.25 

1100 

1.71808 

0. 

0. 

65 

February  22,  A.M. 

5.65 

8 

57 

1.00 

9 

7 

47.00 

646.00 

1000 

1.54799 

0. 

0. 

66 

ft                  «            U 

35.50 

35.75 

it 

9 

15 

39.00 

9 

21 

12.00 

333.00 

450 

1.35135 

316.03 

29054.7 

67 

U                  <£            U 

35.50 

36.50 

tt 

9 

21 

51.25 

9 

28 

0.50 

369.25 

450 

1.21869 

519.69 

43087.9 

68 

U                   (C            11 

35.50 

36.75 

u 

9 

28 

45.00 

9 

36 

26.00 

461.00 

500 

1.08460 

720.20 

53142.4 

69 

((                   U              ( 

35.50 

37.25 

tt 

9 

36 

26.00 

9 

43 

4.50 

398.50 

400 

1.00376 

832.26 

56834.2 

70 

U                   tt             t 

35.75 

37.25 

tt 

9 

43 

57.00 

9 

50 

16.00 

379.00 

350 

0.92348 

934.74 

58727.1 

71 

tt                  tt             t 

35.75 

37.25 

it 

9 

51 

14.00 

9 

58 

9.00 

415.00 

350 

0.84337 

1033.30 

59287.8 

72 

tt                  t(             t 

36.75 

37.25 

tt 

9 

59 

12.50 

10 

7 

48.75 

516.25 

400 

0.77482 

1115.02 

58776.2 

73 

tt                  tt             t 

39.00 

35.75 

u 

10 

21 

10.00 

10 

30 

50.25 

580.25 

450 

0.77553 

1115.02 

58830.1 

74 

tt                   It             t 

38.25 

36.00 

it 

10 

42 

44.50 

10 

51 

14.50 

510.00 

350 

0.68627 

1204.84 

56253.1 

75 

tt                  tt             t 

38.25 

36.25 

u 

10 

58 

28.00 

11 

6 

35.00 

487.00 

300 

0.61602 

1277.98 

53559.4 

76 

tt                  tt             t 

38.25 

36.25 

it 

11 

13 

59.00 

11 

21 

17.00 

438.00 

200 

0.45662 

1482.56 

46056.1 

77 

February  22,  A.M. 

37.50 

35.75 

9.96 

11 

33 

20.00 

11 

40 

9.50 

409.50 

350 

0.85470 

1482.56 

86207.6 

78 

tt           tt       tt 

38.00 

36.00 

ft 

11 

46 

21.00 

11 

53 

27.00 

426.00 

350 

0.82160 

1544.87 

86351.4 

79 

"              "      P.M. 

40.75 

35.75 

U 

0 

0 

42.50 

0 

8 

6.00 

443.50 

350 

0.78918 

1604.85 

86164.4 

80 

February  22,  P.M. 

42.75 

36.00 

2.875 

2 

33 

11.00 

2 

38 

31.25 

320.25 

400 

1.24902 

0. 

0. 

81 

ft                  It           U 

43.75 

36.50 

tt 

2 

42 

3.00 

2 

47 

52.00 

349.00 

400 

1.14613 

118.59 

9247.0 

82 

tt              tt         tt 

44.00 

37.00 

U 

2 

48 

41.00 

2 

54 

45.50 

364.50 

350 

0.96022 

325.39 

21256.6 

83 

tl                   U           tt 

44.25 

37.25 

it 

2 

55 

46.00 

3 

2 

14.50 

388.50 

300 

0.77220 

519.86 

27310.9 

84 

tt              tt         tt 

43.75 

37.25 

tt 

3 

2 

14.50 

3 

9 

41.00 

446.50 

300 

0.67189 

612.22 

27985.1 

85 

It            tl        tt 

43.50 

37.00 

It 

3 

11 

21.50 

3 

18 

48.00 

446.50 

250 

0.55991 

704.44 

26833.8 

86 

ft            If        tl 

43.50 

36.75 

It 

3 

20 

34.50 

3 

27 

47.00 

432.50 

200 

0.46243 

777.58 

24462.9 

87 

It                   U           tt 

43.25 

36.50 

It 

3 

33 

10.50 

3 

44 

17.00 

666.50 

200 

0.30007 

882.02 

18006.4  i 

88 

tt              tt         tl 

42.25 

36.25 

tt 

4 

2 

0. 

0 

0. 

1195.06 

0. 

89 

It                  11            U 

It 

4 

10 

0. 

0 

0. 

1054.25 

0. 

90 

February  22,  P.M. 

42.00 

37.75 

1.00 

4 

27 

3.50 

4 

33 

47.00 

403.50 

250 

0.61958 

118.59 

4998.8 

91 

a           it       a 

a 

4 

35 

1.00 

4 

42 

16.50 

435.50 

300 

0.68886 

73.14 

3427.7 

92 

it           tt       11 

41.00 

37.25 

a 

4 

45 

42.00 

5 

2 

54.00 

1032.00 

400 

0.38760 

296.39 

7815.6 

EXPEEIMENTS   UPON  THE   TREMONT  TURBINE. 


35 


II.  —  CONTINUED. 

TREMONT  MILLS,  IN  LOWELL,  MASSACHUSETTS. 


11 

12 

13 

14 

15 

10 

17 

18 

19 

20 

21 

22 

23 

24 

Height 
of  the 

Height 
of  the 

Total 

Quantity 

Total  power 

Ratio 

Velocity 

Velocity 

Ratio 
of  the 

Quantity  of 

Ratio 
of  the 

Direction 

Mean 

No. 

water 

water 
after 

fall 

Depth  of 

of  water 

of  the  water 

of  the 

due  to  the 

of  the 

velocity  of 
the  interior 

passed  the 

reduced 

of  the 
water 

elevation 

of  the 

above  the 

passing 

water 

I  '   h 

in  pounds 

fall  acting 

interior 

circumrnce 

wheel, 

quantity  in 

leaving 

of  the 

experi- 
ment. 

wheel, 
taken  in 
the 

the 
wheel, 
taken 
in  the 

acting 
upon  th 
wheel, 

on  the 
weir, 

the  weir,  in 
cubic  feet 

avoirdupois 
raised  one 

useful 
effect  to 
the  power 

on  the 
wheel,  in 

circumfnce 
of  the 
wheel,  in 

of  the 
wheel  to 
the 
velocity 

reduced  to  a 
uniform  fall 
of  13  feet,  in 

column  21 
to  the 
reduced 

the  wheel, 
as 
indicated 
by  the 

pointer  on 
the  bell 

forebay, 

wheel- 

in  feet. 

in  feet. 

per  second. 

foot  per 

expended. 

feet  per 

feet  per 

due  to  the 

cubic  feet 

quantity  in 

vane. 

crank. 

in  feet. 

in,  feet 

second. 

second. 

second. 

.     ° 

per  second. 

experiment 

wheel. 

30. 

deg. 

m. 

Feet. 

51 

15.095 

2.337 

12.758 

1.9173 

143.3319 

114060.7 

0.34589 

28.6468 

31.4548 

1.09802 

144.6849 

1.04309 

12 

0 

-  -0.002 

52 

15.128 

2.255 

12.873 

1.8792 

139.2094 

111778.7 

0.59379 

28.7756 

26.6739 

0.92696 

139.8945 

1.00856 

17 

82 

-  -0.001 

53 

15.134 

2.225 

12.909 

1.865G 

137.7518 

110917.6 

0.67462 

28.8159 

24.2121 

0.84023 

138.2365 

0.99660 

22 

19 

-  -0.007 

54 

15.134 

2.194 

12.940 

1.8660 

137.7962 

111219.8 

0.70592 

28.8504 

22.8914 

0.79345 

138.1153 

0.995.73 

25 

0 

-  -0.013 

55 

15.139 

2.189 

12.950 

1.858b 

137.0026 

110664.7 

0.72485 

28.8616 

21.7272 

0.75281 

137.2668 

0.98961 

28 

56 

—0.011 

56 

15.138 

2.187 

12.951 

1.8450 

135.5434 

109494.5 

0.74499 

28.8627 

20.4557 

0.70873 

135.7996 

0.97903 

33  58 

-^0.016 

57 

15.143 

2.178 

12.965 

1.8408 

135.0974 

109252.2 

0.74975 

28.8783 

19.7365 

0.68344 

135.2797 

0.97529 

37  47 

—0.014 

58 

15.144 

2.168 

12.976 

1.8336 

134.3304 

108724.1 

0.75772 

28.8905 

19.0852 

0.66060 

134.4546 

0.96934 

42 

43 

—0.011 

59 

15.151 

2.152 

12.999 

1.8240 

133.3014 

108082.5 

0.76049 

28.9161 

18.3511 

0.63463 

133.3066 

0.96106 

47 

;;o 

—0.005 

60 

15.153 

2.139 

13.014 

1.8186 

132.7344 

107746.9 

0.76158 

28.9328 

17.7125 

0.61219 

132.6630 

0.95642 

54 

:{? 

—0.004 

61 

15.155 

2.129 

13.026 

1.8117 

131.9960 

107246.3 

0.76260 

28.9461 

17.0556 

0.58922 

131.8642 

0.95066 

59 

;!'.) 

—0.042 

62 

15.162 

2.122 

13.040 

1.8022 

130.9913 

106544.4 

0.76363 

28.9617 

16.3751 

0.56541 

130.7903 

0.94292 

76 

:!(> 

—0.021 

63 

15.162 

2.134 

13.028 

1.8013 

130.8932 

106366.6 

0.75012 

28.9484 

15.0091 

0.51848 

130.7525 

0.94265 

94 

•I 

—0.022 

64 

15.079 

2.359 

12.720 

1.9742 

149.5470 

118652.1 

0. 

28.6041 

36.4332 

1.27370 

151.1840 

1.08995 

65 

15.139 

1.969 

13.170 

1.7160 

121.9685 

100194.6 

0. 

29.1057 

32.8262 

1.12783 

121.1788 

0.87363 

66  115.148 

2.071 

13.077 

1.6829 

118.5511 

96699.5 

0.30046 

29.0028 

28.6564 

0.98806 

118.2016 

0.85216 

6 

30 

4-0.001 

67 

15.159 

2.025 

13.134 

1.6590 

116.0987 

95112.0 

0.45302 

29.0659 

25.8432 

0.88912 

115.5050 

0.83272 

8 

80 

+0.002 

68 

15.164 

1.988 

13.176 

1.6409 

114.2599 

93904.9 

0.56592 

29.1123 

22.9997 

0.79003 

113.4942 

0.81823 

12 

82 

—0.020 

69 

15.171 

1.956 

13.215 

1.6309 

113.2448 

93346.0 

0.60885 

29.1554 

21.2856 

0.73007 

112.3198 

0.80976 

16 

5 

—0.027 

70 

15.173 

1.920 

13.253 

1.6139 

111.5197 

92188.4 

0.63703 

29.1973 

19.5831 

0.67072 

110.4502 

0.79628 

21 

17 

—0.006 

71 

15.179 

1.897 

13.282 

1.5959 

109.7130 

90893.3 

0.65228 

29.2292 

17.8844 

0.61187 

108.5420 

0.78252 

29 

56 

—0.014 

72 

15.183 

1.872 

13.311 

1.5793 

108.0452 

89707.1 

0.65520 

29.2611 

16.4306 

0.56152 

106.7756 

0.76979 

39 

2 

4-0.001 

73 

15.179 

1.869 

13.310 

1.5783 

107.9493 

89620.7 

0.65643 

29.2600 

16.4457 

0.56205 

106.6848 

0.76913 

40 

18 

—0.029 

74 

15.159 

1.833 

13.326 

1.5541 

105.5341 

87720.9 

0.64127 

29.2776 

14.5530 

0.49707 

104.2353 

0.75147 

69 

27 

—0.028 

75 

15.174 

1.812 

13.362 

1.5371 

103.8516 

86555.6 

0.61879 

29.3171 

13.0631 

0.44558 

102.4352 

0.73850 

95 

0 

—0.024 

76 

15.183 

1.771 

13.412 

1.5034 

100.5410 

84110.0 

0.54757 

29.3719 

9.6830 

0.32966 

98.9847 

0.71362 

144 

56 

—0.029 

77 

15.079 

2.196 

12.883 

1.8620 

137.3618 

110380.8 

0.78100 

28.7868 

18.1246 

0.62961 

137.9842 

0.99478 

55 

52 

—0.037 

78 

15.079 

2.183 

12.896 

1.8583 

136.9694 

110176.6 

0.78375 

28.8013 

17.4226 

0.60492 

137.5206 

0.99144 

64 

0 

—0.029 

79 

15.087 

2.175 

12.912 

1.8544 

136.5469 

109973.0 

0.78350 

28.8192 

16.7351 

0.58069 

137.0114 

0.98777 

74 

28 

—0.018 

80 

14.774 

1.427 

13.347 

1.2914 

80.4534 

66979.0 

0. 

29.3006 

26.4865 

0.90396 

79.4007 

0.57243 

0 

30 

81    14.769 

1.400 

13.369 

1.2737 

78.8433 

65746.8 

0.14065 

29.3248 

24.3046 

0.82881 

77.7476 

0.56051 

1 

30 

+0.008 

82    14.772 

1.377 

13.395 

1.2492 

76.6213 

64018.1 

0.33204 

29.3.533 

20.3622 

0.69369 

75.4831 

0.54419 

4 

32 

+0.010 

83  H4.783 

1.348 

13.435 

1.2206 

74.0590 

62062.0 

0.44006 

29.3971 

16.3751 

0.55703 

72.8501 

0.52521 

11 

80 

+0.002 

84    14.793 

1.315 

13.478 

1.1960 

71.8750 

60424.6 

0.46314 

29.4441 

14.2480 

0.48390 

70.5889 

0.50890 

20 

9 

—0.001 

85    14.806 

1.293 

13.513 

1.1748 

70.0063 

59006.4 

0.45476 

29.4823 

11.8733 

0.40273 

68.6646 

0.49503 

41 

84 

—0.022 

86    14.820 

1.264 

13.556 

1.1497 

67.8158 

57342.0 

0.42661 

29.5292 

9.8061 

0.33208 

66.4105 

0.47878 

81 

40 

—0.025 

87    14.803 

1.244 

13.559 

1.1113 

64.5053 

54554.9 

0.33006 

29.5324 

6.3633 

0.21547 

63.1616 

0.45536 

—0.026 

88    14.762 

1.246 

13.516 

1.0623 

60.3593 

50886.6 

0. 

29.4856 

0. 

0. 

59.1959 

0.42677 

89    14.771 

1.240 

13.531 

1.0630 

60.4190 

50993.4 

0. 

29.5020 

0. 

0. 

59.2216 

0.42695 

90 

14.806 

0.821 

13.985 

0.7798 

38.2210 

33340.8 

0.14993 

29.9928 

13.1386 

0.43806 

36.8505 

0.26567 

+0.004 

91    14.815 

0.814 

14.001 

0.7846 

38.5699 

33683.5 

0.10176 

30.0099 

14.6079 

0.48677 

37.1655 

0.26794 

+0.030 

92  i  14.832 

0.812 

14.020 

0.7653 

37.1733 

32508.0 

0.24042 

30.0303 

8.2193 

0.27370 

35.7956 

0.25806 

—0.006 

36 


EXPERIMENTS  UPON  THE  TEEMONT  TURBINE 


DESCRIPTION  OF  THE  DIAGRAM  REPRESENTING  THE  EXPERIMENTS. 

75.  For  the   purpose   of   presenting    a  general  view   of  the   experiments,   the 
coefficients  of  effect,  at   different  velocities,  are   plotted   at  figure   1,  plate  VI.,  on  a 
system   of  coordinates.      The   ratios   of    the   velocities   of  the   interior   circumference 
of   the   wheel,  to    the   velocities    due    the    fall    acting   upon    the   wheel,   given    in 
column   20,  table   II.,  are   taken   to   represent  the   velocities;    these   ratios  are  here 
called  the   velocities,  and  are  taken  on  the  axis  of  abscissas  AX;    the  correspond- 
ing coefficients   of  effect   given  in   column   17,  table   II.,   are   taken   upon    the   axis 
of  ordinates  AY.  » 

76.  The   line   CD  represents  the   experiments  made   with   the   regulating   gate 
fully   raised ;  —  to   avoid   confusion   a    portion    of   the    experiments    are   omitted ;  — 
the    experiments   represented    are   those    numbered   from   4   to   42,   inclusive,   which 
were    made    in    regular    sequence,  with   gradually  increasing  weights.      It  will    be 
observed   in   the   table  of  experiments,  that  several  trials  were  made  with  the  brake 
entirely   removed;    these   were  made,  generally,  after  the   wheel   had  been   left  for 
some   time,  for  the  purpose   of  seeing  if  it  was  in   as   good  running  order  as  usual ; 
if  any  material  change  had   taken  place,  it  would  have  been  indicated  by  a  change 
in  the  velocity  of  the  wheel. 

The   experiments  thus   made,  omitting  experiment   12,  in  which  the   height  in 
the   wheelpit   was  not   observed,  are   collected   together  in   the   following  table. 


Number  of  the 
experiment. 

Ratio   of  the  velocity  of  the  interior  cir- 
cumference of  the  wheel,  to  the  velocity 
due  the  fall  acting  upon  the  wheel. 

13 
23 
31 

38 
45 

1.33366 
1.33567 
1.33635 
1.33544 
1.33521 

Mean     .     .     . 

1.33527 

The   greatest  variation  in  these  velocities  is  in   experiment   13,   which  is 
part  below   the   mean ;     the   running   condition    of   the   wheel    must,    consequently, 
have   been   nearly  uniform. 

In  all  the  experiments  with  the  brake  removed,  the  coefficient  of  effect,  of 
course,  is  nothing,  and  they  would  be  represented  on  the  diagram  by  points  on 
the  axis'  of  abscissas;  for  the  sake  of  distinctness,  only  one  of  those  tried  when 
Ihe  gate  was  at  its  full  height,  is  represented  on  the  diagram. 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE.  37 

There  is  a  small  irregularity  in  the  line  CD,  at  numbers  26  and  27;  both 
these  experiments  were  made  with  the  same  weight  in  the  scale,  and  under  sim- 
ilar circumstances,  except  that  in  26,  water  was  used  to  lubricate  the  friction 
pulley,  and  in  27  oil  was  used. 

It  has  been  stated,  that,  with  heavy  loads,  the  brake  operates  much  more 
steadily  with  oil  as  a  lubricator,  than  with  water,  and  the  change  in  the  lubrica- 
tor at  experiment  27,  was  made  in  consequence  of  the  difficulty  experienced  by 
the  operator,  in  regulating  the  tension  of  the  brake  screws.  In  experiment  26, 
nearly  his  whole  strength,  applied  to  the  extremity  of  a  wrench  about  three  feet 
long,  was  required  to  move  the  nuts,  whereas,  in  experiment  27,  the  same  opera- 
tion was  performed  with  great  ease.  Experiment  26  was  of  much  shorter  dura- 
tion than  experiment  27,  and  a  portion  of  the  discrepancy  may  be  due  to  a 
proportionally  less  perfect  observation  of  the  data  in  26. 

The  line  CD  shows  that,  with  a  velocity  of  the  interior  circumference  of  the 
wheel  not  less  than  44  or  more  than  75  per  cent,  of  that  due  to  the  fall,  the 
useful  effect  is  75  per  cent,  or  more,  of  the  total  power  expended.  Beyond 
these  points,  the  change  in  the  coefficient  of  effect  is  nearly  equal  for  equal  and 
opposite  variations  of  speed ;  thus,  the  diagram  indicates  that  the  coefficient  of 
effect  is  70  per  cent,  of  the  power  expended,  at  the  velocities  0.360  and  0.834. 

0.436  —  0.360  =  0.076 
0.834  —  0.750  =  0.084. 

Taking  the  mean  of  the  extreme  velocities,  that  is,  of  0,  when  the  wheel 
was  still,  and  1.335,  when  the  brake  was  removed,  we  have 

1.335  +  0  =  0j66m 
SB 

which  is  not  far  from  the  velocity  giving  the  maximum  coefficient  of  effect; 
that  is  to  say,  when  the  gate  is  fully  raised,  the  coefficient  of  effect  is  a  maximum 
when  the  wheel  is  moving  with  about  half  its  maximum  velocity, 

77.  Experiments  43  and  44  were  both  made  with  the  gate  fully  raised,  but 
the  wheel  at  rest,  the  brake  being  screwed  up  sufficiently  tight  to  prevent  the 
wheel  from  revolving ;  —  they  were  made  for  the  purpose  of  ascertaining  the  total 
effort  that  could  be  exercised  by  the  wheel. 

By  reference  to  column  9,  of  the  table  of  experiments,  it  will  be  seen  that, 
in  experiment  43,  the  weight  sustained  was  4213.38  pounds,  and  in  44,  the  weight 
was  3946.38  pounds.  These  experiments  were  made  under  circumstances  nearly 
identical,  except  that  in  43,  the  weight  preponderated,  and  in  44,  the  power  of 


38  EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 

the  wheel  preponderated.  In  43,  the  weight  was  the  least  that  would  cause  the 
scale  to  lower  when  the  bell  crank  was  placed  horizontally,  and  then  left  free ; 
on  the  other  hand,  in  experiment  44,  the  weight  was  the  greatest  that  would 
allow  the  scale  to  be  raised  under  the  same  circumstances;  that  is  to  say,  in  43, 
the  weight  represents  the  force  exercised  by  the  water  against  the  wheel,  plus 
the  friction  of  the  entire  apparatus,  and  in  44,  the  weight  represents  the  same 
thing,  minus  the  friction;  the  difference  of  the  weights,  or  4213.38  —  3946.38  =  267 
pounds,  represents  double  the  friction,  and  the  true  force  exercised  by  the  water 
against  the  wheel,  is  represented  by  the  weight 

4213.38  +  3946.38  __  ^Q  gg  pounc}s 
2 

This  weight  acted  at  a  distance  from  the  centre  of  the  wheel,  equal  to  the 
effective  length  of  the  brake,  or  10.827778  feet  (art.  50). 

The  radius  of  the  turbine,  at  the  outer  extremities  of  the  buckets,  is  4.146 
feet  (art.  35),  consequently,  the  equivalent  force  acting  tangentially  at  the  outer 
extremities  of  the  buckets,  was 

4079.88  X  10.827778        ^^  dg 

4.146 

78.  The   line   E  F  represents   the   experiments  numbered  77,  78,  and  79,  made 
with  the   gate   raised   9.96   inches,   or   about   87   per   cent,   of  the    full   height.      By 
a  reference   to   the   table    of  experiments,  it  will   be    seen  that,  although    the    regu- 
lating  gate   was  lowered    13   per   cent.,  the   quantity   of   water  discharged    by   the 
wheel  was  diminished  less  than  one  per  cent. 

79.  The  line  GH  represents   the  experiments  numbered  from  51  to  64,  inclu- 
sive,  made   with  the    gate   raised    8.55   inches,   or   about   three   fourths  of  the    full 
height. 

80.  The   line   IK  represents  the  experiments  numbered  from  65  to  76,  inclu- 
sive, made  with  the  gate   raised    5.65  inches,  or  nearly  a  half  of  the  full   height. 

81.  The  line  LM  represents  the  experiments  numbered  from  80  to  87,  inclu- 
sive,  made   with   the    gate    raised   2.875    inches,   or    one    fourth   of    its   full    height. 
Experiments   88   and    89    were    made  with    the  same   height   of  gate,  but  with   the 
wheel    held   fast  by  the  brake;    the    force    exerted    by   the   wheel   at  the   distance 
10.827778  feet,  independent  of  friction,  was 

1195.06  -f  1054.25 
-if"         '  = 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE.  39 

82.  The  line  NO  represents  the  three  experiments  numbered  90,  91,  and 
92,  made  with  the  regulating  gate  raised  one  inch. 

An  examination  of  the  diagram  will  show  that  the  velocity  corresponding  to 
the  maximum  coefficient  of  effect,  diminishes  with  the  height  of  the  gate.  For 
heights  not  less  than  one  fourth  of  the  whole  height,  this  diminution  is  sufficiently 
regular ;  for  heights  less  than  one  fourth,  the  experiments  are  not  sufficient  to 
indicate  the  velocity  giving  the  best  effect,  but  the  diminution  is  evidently  more 
rapid  than  for  greater  heights  of  gate. 


PATH  DESCRIBED  BY  A  PARTICLE  OF  WATER  IN  PASSING  THROUGH  THE  WHEEL. 

83.  As  in  many  other  problems  in  hydraulics,  resort  is  here  had  to  a  par- 
ticular hypothesis,  which,  at  best,  is  only  an  approximation  to  the  truth,  neverthe- 
less, it  may  be  the  means  of  throwing  some  light  upon  the  mode  in  which  the 
water  acts  upon  the  wheel. 

The  particular  hypothesis  here  assumed  is  this;  every  particle  of  water  contained 
in  the  wheel,  situated  at  the  same  distance  from  the  axis,  moves  in  the  same  direction  relative 
to  the  radius,  and  writh  the  same  velocity.  According  to  this  hypothesis,  the  successive 
sections  in  which  the  same  particles  of  water  are  found,  are  in  cylindrical  surfaces, 
concentric  with  the  wheel. 

Applying  this  hypothesis  to  experiment  30,  on  the  Tremont  Turbine,  let  us 
suppose 

($'  =  the   mean   quantity   of   water    discharged    through   each   aperture   of   the 

wheel,  in  cubic  feet  per  second. 
to  =  the  angular  velocity  of  the  wheel. 
R  =  the  radius  of  the  circle  inscribing  the  inner  edges  of  the  buckets,  or  0  A, 

figure  3,  plate  VI. 
R  =  the  radius  0  B. 
t  =  the  time  occupied  by  a  particle  of  water  in   passing  from  the  section  A  D 

to   the   section  B  O,  or,  which  is  the   same   thing,  through   the   radial 

distance  R — R. 

A  =  the  area  of  AS  CD,  in  square  feet. 
H=  the  mean   height,  in  feet,  between  the  crowns  of  the  wheel,  between  the 

sections  A  D  and  B  0. 

We  have 

AH=the   volume  of  water  contained  between  the  sections  AD  and  B  C. 


40  EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 

t  is  the  time  occupied  by  a  particle  of  water  in  passing  from  the  section 
AD  to  the  section  B  C,  and  it  will  evidently  be  the  time  required  for  the  dis- 
charge of  the  volume  AH.  We  find  t  by  the  proportion 


If  the  wheel  was  at  rest,  a  particle  of  water  at  A  would  arrive  at  B  in  the 
time  t,  but  the  wheel  is  moving  with  the  angular  velocity  to,  therefore  the  point 
B,  in  the  time  t,  will  have  advanced  to  E,  and 


consequently,  a  particle  of  water  at  A,  instead  of  being  at  B,  at  the  end  of 
the  time  t,  will  have  arrived,  by  some  path,  at  the  point  E.  In  this  manner, 
by  taking  successive  values  of  R,  sufficiently  near  to  each  other,  the  entire  path 
of  a  particle  of  water,  from  its  entrance  into  the  wheel,  up  to  the  moment  of 
its  discharge,  may  be  traced;  and  as,  by  the  hypothesis,  all  the  particles  at  the 
same  distance  from  the  axis  move  with  the  same  velocity,  and  in  the  same 
relative  direction,  the  path  of  the  entire  stream,  from  its  entrance  into  the  wheel 
to  its  discharge,  will  be  determined. 

In   experiment   30,  we  have  the  total  quantity  discharged   by  the  wheel  equal 
to   138.1892  cubic  feet  per  second;    as  the  wheel  has  forty-four  apertures, 

q,_  138.1892  _314066  cubic  feet  per  gecond. 

44 

The   velocity  of  the   interior  circumference   of  the  wheel  was   18.0474  feet  per 
second,  and  the  interior  radius  of  the  wheel  being  3.375  feet,  we  have 


0,  =  ^^^=  5.3474  feet  per  second, 

o.o/O 

consequently, 


BE_wuiKAu.  _  j  702g  RAH 
3.14066 

84.  The  successive  steps  in  the  calculation  for  the  entire  path,  are  given  in 
table  III. 

The  arcs  of  circles  F  G,  HI,  etc.  are  drawn  on  a  plan  of  the  buckets,  figure 
2,  plate  VI.,  with  the  radii  contained  in  the  first  column. 

COLUMN  2  contains  the  entire  areas  of  these  circles. 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 


41 


COLUMN  3  contains  the  areas  of  the  rings  comprised  between  these  circles, 
which  are  obtained  by  taking  the  differences  of  the  successive  areas  in  column  2. 

COLUMN  4  contains  the  areas  reduced  to  square  feet,  of  that  part  of  each  ring 
corresponding  to  a  single  aperture  in  the  wheel,  including  also  the  area  occupied 
by  the  thickness  of  the  corresponding  part  of  one  bucket. 

COLUMN  5.  Corrections  for  the  thickness  of  the  buckets ;  —  these  are  deduced 
from  measurements  taken  on  a  full  sized  plan  of  the  buckets. 

COLUMN  6.  True  areas  of  the  partial  rings,  being  the  differences  of  the  cor- 
responding areas  in  columns  4  and  5. 

COLUMN  7.  Mean  heights  of  the  partial  rings;  —  these  are  also  taken  from  a 
full  sized  drawing  of  the  wheel. 

COLUMN  8.  Volumes  of  the  partial  rings,  or  the  products  of  the  corresponding 
numbers  in  columns  6  and  7. 

COLUMN  9.  Volumes  between  the  radius  R,  and  the  successive  values  of  the 
radius  tf.  These  are  obtained  by  adding  together  the  volumes  of  the  partial 
rings,  up  to  the  corresponding  radius ;  —  they  are  the  successive  values  of  A  H. 

COLUMN  10.     The  ordinates;  —  these  are  successive  values  of 

1.7026  RAH, 

the  successive  values  of  K  being  taken  in  feet,  instead  of  inches,  as  they  are  given 
in  column  1. 


TABLE  III. 


1 

a 

3 

4 

5 

6 

,7 

8 

9 

10 

Value  of 
K,  and 
successive 
values  of 
R'.  Inches. 

Areas  in  square 
inches,  of  circles 
of  the  radii  in 
the  last  column. 

Areas  in 
square  inches 
of  the 
complete  rings. 

A 

of  the  areas 
of  the  rings 
in  the  last 
column. 
Square  feet. 

Correction  for 
the  thickness 
of  the  bucket, 
in  square  feet. 

True  areas  of 
the  partial 
rings,  in 
square  feet. 

Mean  height 
of  the 
partial  rings, 
infect. 

Volumes  of  the 
partial  rings, 
in  cubic 
feet. 

Volumes 
between  K  and 
the  successive 
values  of  Rf. 
Cubic  feet. 

Ordinates  in 
feet,  to  be 
measured  on 
arcs  of  the 
corresponding 
radii  hi 
column  1. 

40.5 

5152.997 

41.5 

5410.G08 

257.611 

0.04066 

0.00091 

0.03975 

0.9264 

0.03682 

0.03682 

0.2168 

42.5 

5674.502 

263.894 

0.04165 

0.00099 

0.04066 

0.9080 

0.03692 

0.07374 

0.4447 

43.5 

5944.679 

270.177 

0.04264 

0.00106 

0.04158 

0.8940 

0.03717 

0.11091 

0.6845 

44.5 

6221.139 

276.460 

0.04363 

0.00115 

0.04248 

0.8840 

0.03755 

0.14846 

0.9373 

45.5 

6503.882 

282.743 

0.04462 

0.00128 

0.04334 

0.8775 

0.03803 

0.18649 

1.2039 

46.5 

6792.909 

289.027 

0.04562 

0.00146 

0.04416 

0.8755 

0.03866 

0.22515 

1.4854 

47.5 

7088.218 

295.309 

0.04661 

0.00174 

0.04487 

0.8800 

0.03949 

0.26464 

1.7835 

48.5 

7389.811 

301.593 

0.047'60 

0.00212 

0.04548 

0.8920 

0.04057 

0.30521 

2.1003 

49.0 

7542.964 

153.153 

0.02417 

0.00138 

0.02279 

0.9055 

0.02064 

0.32585 

2.2654 

49.25 

7620.129 

77.165 

0.01218 

0.00078 

0.01140 

0.9145 

0.01042 

0.33627 

2.3498 

49.50 

7697.687 

77.558 

0.01224 

0.00087 

0.01137 

0.9210 

0.01047 

0.34674 

2.4352 

49.75 

7775.638 

77.951 

0.01230 

0.00081 

0.01149 

0.9277 

0.01066 

0.35740 

2.5228 

6 


42  EXPERIMENTS  UPON  THE  TREMONT  TURBINE. 

85.  The  arcs  FG,  HI,  etc.,  figure   2,  plate  VI,  are   taken   equal   to  the   ordi- 
nates   0.2168,  0.4447  etc.,  in   column   10   of  the   table ;    the   points    Q,  G,  I,  etc.  K, 
are    joined   by   a   line,  which   is   the   limit   of  the   stream   on   one   side.      The   limit 
on    the    other    side    is    found    by  making    the    arcs   GL  =  FN,  IM=HO,  etc.; 
the   points  R,  L,  M,  etc.  P,  being  joined   by  a  line,  give   the   limits  of  the   stream 
on   this   side. 

86.  By  an  inspection   of   the  figure,  it  is  plain  that,   in   experiment   30,  the 
path   of  the   water    through    the   wheel    must    have    been    a    continuation    of    the 
direction    given    to    it    by   the    fixed    guides    VW,   and   that   there   was   no   sudden 
change    of    direction   or   velocity,   up    to    a    point    near   where    the   water   was   dis- 
charged  from    the   wheel.      The    abrupt    change    at    this    point,   indicated    by   the 
figure,   could    not,    in    reality,   have    taken    place,   as    we    know    by   the    direction 
assumed  by  the  vane,  which  is  represented  at  ST  in  its  mean  position  during  the 
experiment. 

87.  The  foregoing  hypothesis  will   evidently  lead   to   results  more   nearly  cor- 
rect, the   nearer  the   buckets    are  to   each   other,  until,  in    the   case   in   which  the 
spaces    between   them    are    infinitely   small,  it  will    give   the   path   accurately.       In 
applications  like   the   above,  where   the   spaces   are   very   considerable,  it  is   assumed 
by   the   hypothesis  that  the  water   passes  through    in   curved   laminse,    superimposed 
on   each  other,  the   first   of  which,  in   contact  with  the   concavity   of   the   bucket, 
is   constrained    by   it   and    the    rotation    of    the   wheel,  to    move    in    a    particular 
path;     this,  in    its    turn,   constrains  the    next    lamina  to   move   in   a  similar  path; 
and   so   on. 

By  an  inspection  of  figure  2,  plate  VI.,  it  is  reasonable  to  suppose,  that  a 
lamina,  far  removed  from  the  concavity  of  the  bucket,  will  take  a  path  differing 
from  that  of  a  lamina  near  it ;  the  abruptness  in  the  curve  near  its  extremity, 
will  be  diminished,  somewhat  in  proportion  to  the  distance  of  the  lamina  from 
the  concavity  of  the  bucket,  the  water  passing  out  from  the  wheel  more 
nearly  in  the  direction  in  which  it  was  moving,  during  its  approach  to  the 
circumference  of  the  wheel.  These  views  go  far  to  explain  the  discrepancy 
between  the  path  determined  by  the  hypothesis,  and  the  direction  assumed  by 
the  vane. 

88.  Whatever    objection    may  be    made    to    the    method   by  which  the   path, 
given   in   figure   2,   plate   VI.,   is   obtained,   it    cannot  be    denied    that    its    general 
course   must  have    been  nearly   as   represented ;    this   being   admitted,   it   is   difficult 
to   see   how   centrifugal    force    can    operate    in   the   important   manner   that  is   com- 
monly   assigned   to   it.      The    path    is    concave    to   the    axis   only   in   a   very   slight 
degree,   and   through   a   part   only   of   its   course ;    nevertheless,   it    is   only   in   con- 


EXPERIMENTS  UPON  THE  TREMONT  TURBINE.  43 

sequence  of  a  concavity  in  the  path,  that  centrifugal  force  can  have  any  exist- 
ence. With  the  gate  only  partially  raised,  this  force  may  act  powerfully  in 
increasing  the  discharge,  and  a  similar  effect  may  be  produced,  at  high  velocities, 
with  the  gate  fully  raised ;  but  in  experiment  30,  giving  the  maximum  coefficient 
of  effect,  it  can  have  had  only  a  slight  action. 


RULES  FOR  PROPORTIONING  TURBINES, 


89.  IN  making  the    designs  for  the   Tremont,   and   other  turbines,  the   author 
has    been    guided    by   the   following  rules,  which   he   has  been    led   to   by   a   com- 
parison  of   several    turbines  designed    by   Mr.   Boyden,  which   have   been   carefully 
tested  and  found  to  operate  well. 

Rule  1st.  The  sum  of  the  shortest  distances  between  the  buckets,  should  be 
equal  to  the  diameter  of  the  wheel. 

Rule  2d.  The  height  of  the  orifices  at  the  circumference  of  the  wheel,  should 
be  equal  to  one  tenth  of  the  diameter  of  the  wheel. 

Rule  3d.  The  width  of  the  crowns  should  be  four  times  the  shortest  dis- 
tance between  the  buckets. 

Rule  4th.  The  sum  of  the  shortest  distances  between  the  curved  guides,  taken 
near  the  wheel,  should  be  equal  to  the  interior  diameter  of  the  wheel. 

The  turbines,  from  a  comparison  of  which  the  above  rules  were  derived, 
varied  in  diameter  from  twenty-eight  inches  to  nearly  one  hundred  inches,  and 
operated  on  falls  from  thirty  feet  to  thirteen  feet.  The  author  believes  that 
they  may  be  safely  followed  for  all  falls  between  five  feet  and  forty  feet,  and 
for  all  diameters  not  less  than  two  feet,  and,  with  judicious  arrangements  in  other 
respects,  and  careful  workmanship,  a  useful  effect  of  seventy-five  per  cent,  of  the 
power  expended,  may  be  relied  upon.  For  falls  greater  than  forty  feet,  the 
second  rule  should  be  modified,  by  making  the  height  of  the  orifices  smaller  in 
proportion  to  the  diameter  of  the  wheel. 

90.  Taking  the    foregoing    rules  as   a  basis,  we  may,  by   aid   of    the   experi- 
ments on  the   Tremont  Turbine,  establish  the  following  formulas. 

Let  D  =  the  diameter  of  the  wheel  at  the  outer  extremities  of  the  buckets. 
d=fhe  diameter  of  the  wheel,  at  the  interior  extremities  of  the  buckets. 
-£T=the  height   of   the    orifices  of  discharge,   at  the   outer   extremities   of 

the   buckets. 
W=  the  width  of  the  crowns  occupied  by  the  buckets. 


RULES  FOR  PROPORTIONING  TURBINES.  45 

N=  the  number  of  buckets. 
n  =  the  number  of  guides. 
P  —  the   horse-power    of    the   turbine  ;     a   horse-power    being    550    pounds 

avoir,  raised  one  foot  per  second. 
h  =  the  fall  acting  upon  the  wheel. 
Q  =  the   quantity   of   water   expended    by  the   turbine,  in   cubic    feet  per 

second. 

V=  the  velocity  due  the  fall  acting  upon  the  wheel. 

V  =  the  velocity  of  the  water  passing  the  narrowest  sections  of  the  wheel. 
v  =  the  velocity  of  the  interior  circumference  of  the  wheel  :    all  the  veloci- 

ties being  in  feet  per  second. 
C=  the   coefficient   of   V,  or  the   ratio   of   the   real   velocity   of  the   water 

passing    the    narrowest   sections    of   the   wheel,  to   the   theoretical 

velocity  due  the  fall  acting  upon  the  wheel. 

The  unit  of  length  is  the  English  foot. 

It  is  assumed  that  the  useful  effect  is  seventy-five  per  cent,  of  the  total 
power  of  the  water  expended. 

According  to  rule  1,  we  have  the  sum  of  the  widths  of  the  orifices  of  dis- 
charge, equal  to  D.  Then  the  sum  of  the  areas  of  all  the  orifices  of  discharge, 
is  equal  to  DH. 

By  the  fundamental  law  of  hydraulics  we  have 


therefore 


We  can  find  the  value  of  C  in  the  last  equation  by  experiment  30,  on  the 
Tremont  Turbine.  In  that  wheel  we  have  for  the  sum  of  the  widths  of  the 
orifices  of  discharge,  44  X  0.18757  =  8.25308  feet,  and  the  height  of  the  orifices  of 
discharge  =  0.9314  feet.  Then  we  have,  for  the  sum  of  the  areas  of  all  the  ori- 
fices of  discharge, 


=  8.25308  X  0.9314  =  7.68692  square  feet 
By  experiment  30,  we  have 

Q  =  138.1892  cubic  feet  per  second, 
h  =  12.903  feet, 

=  8.0202  feet, 


46  RULES  FOR  PROPORTIONING  TURBINES. 

consequently, 

138.1892  =  7.68692  X  8.0202  V12.903  C, 
or  C=  0.624. 

By  rule  2,  we  have  H—  0.10  D  : 

then  HD=  0.10  D*, 

and  Q  =  JIDVf  =  Q.lQIPC^2gF, 
or  Q  =  0.5  If-  \Th. 

Calling  the  weight  of  a   cubic  foot   of  water   62.33   pounds   avoir,  we   have 

p  _  0.75  X  62-33  o  , 
-550-   -U'1' 

or  P  =  0.085  Qh; 
or,  substituting  the  value  of  Q  just  found, 

P=  0.0425  1?  h  v/T, 
from  which  we  may  deduce 


91.  The  number  of  buckets  is,  to  a  certain  extent,  arbitrary,  and  would  usually 
be  determined  by  practical  considerations  :  some  of  the  ideas  to  be  kept  in  mind 
are  the  following. 

The  pressure  on  each  bucket  is  less,  as  the  number  is  greater  ;  the  greater 
number  will  therefore  permit  of  the  use  of  the  thinner  iron,  which  is  important, 
in  order  to  obtain  the  best  results.  The  width  of  the  crowns  will  be  less  for 
a  greater  number  of  buckets  :  a  narrow  crown  appears  to  be  favorable  to  the 
useful  effect,  when  the  gate  is  only  partially  raised.  As  the  spaces  between  the 
buckets  must  be  proportionally  narrower  for  a  larger  number  of  buckets,  the 
liability  to  become  choked  up,  either  with  anchor  ice,  or  other  substances,  is 
increased.  The  amount  of  power  lost  by  the  friction  of  the  water  against  the 
surfaces  of  the  buckets,  will  not  be  materially  changed,  as  the  total  amount  of 
rubbing  surface  on  the  buckets,  will  be  nearly  constant  for  the  same  diameter: 
there  will  be  a  little  less  on  the  crown,  for  the  larger  number.  The  cost  of 
the  wheel  will  probably  increase  with  the  number  of  buckets.  The  thickness 
and  quality  of  the  iron,  or  other  metal  intended  to  be  used  for  the  buckets, 
will  sometimes  be  an  element.  In  some  waters,  wrought  iron  is  rapidly  corroded. 


RULES   FOR  PROPORTIONING   TURBINES.  47 

The  author  is  of  opinion  that  a  general  rule  cannot  be  given  for  the  num- 
ber of  buckets;  among  the  numerous  turbines  working  satisfactorily  in  Lowell, 
there  are  examples  in  which  the  shortest  distance  between  the  buckets  ia  as 
small  as  0.75  inches,  and  in  others  as  large  as  2.75  inches. 

As  a  guide  in  practice,  to  be  controlled  by  particular  circumstances,  the  fol- 
lowing is  proposed  ;  to  be  limited  to  diameters  of  not  less  than  two  feet  ; 


Taking  the  nearest  whole  number  for  the  value  of  JV! 

The  Tremont  Turbine  is  8£  feet  in  diameter,  and,  according  to  the  proposed 
rule,  should  have  fifty-five  buckets,  instead  of  forty-four.  With  fifty-five  buckets, 
the  crowns  should  have  a  width  of  7.2  inches,  instead  of  9  inches  ;  with  the 
narrower  width,  it  is  probable  that  the  useful  effect,  in  proportion  to  the  power 
expended,  would  have  been  a  little  greater  when  the  gate  was  partially  raised. 

92.     By  the  3d  rule,  we  have  for  the  width  of  the  crowns, 


"  N 

and  for  the  interior  diameter  of  the  wheel 


By  the  4th  rule,  d  is  also  equal   to  the  sum   of  the  shortest  distances  between 
the  guides,  where  the  water  leaves  them. 

93.  The    number   n,   of  the    guides,   is,   to    a    certain    extent,    arbitrary;     the 
practice   at   Lowell    has    been,  usually,    to   have    from    a  half   to   three   fourths   of 
the   number   of    the   buckets;    exactly   half  would  probably   be   objectionable,   as  it 
would  tend  to  produce  pulsations,  or  vibrations. 

94.  The   proper  velocity  to   be   given   to   the   wheel,   is   an    important   consid- 
eration.     Experiment   30,  on   the   Tremont   Turbine,  gives  the   maximum  coefficient 
of  effect  for   that  wheel;    in   that   experiment  the  velocity  of  the   interior   circum- 
ference  of  the  wheel,  is   0.62645   of  the  velocity  due   to   the   fall   acting   upon   the 
wheel.      By  reference   to   the   other   experiments  with  the   gate   fully  raised,  it  will 
be   seen,   however,    that   the   coefficient    of    effect  varies    only  about  two   per  cent. 
from   the   maximum,   for   any  velocity    of  the   interior   circumference,   between   fifty 
per  cent,   and   seventy  per   cent,   of  that   due   to   the   fall   acting    upon   the   wheel. 
By   reference    to    the    experiments    in   which    the    gate   is   only   partially   raised,   it 
will   be   seen   that  the  maximum   corresponds  to  slower  velocities;    and  as  turbines, 


48 


RULES  FOR  PROPORTIONING  TURBINES. 


to  admit  of  being  regulated  in  velocity  for  variable  work,  must,  almost  necessarily, 
be  used  with  a  gate  not  fully  raised,  it  would  appear  proper  to  give  them  a 
velocity  such,  that  they  will  give  a  good  effect  under  these  circumstances. 

With  this  view,  the  following  is  extracted  from  the  experiments  in  table  II. 


Ratio   of  the  velocity  of  the  interior  cir- 

Number of  the 
experiment. 

Height  of  the  regulat- 
ing gate,  in  inches. 

cumference  of  the  wheel,  to  the  velocity 
due  the  fall  acting  upon  the  wheel,  cor- 
responding to  the  maximum  coefficient 

of  effect. 

30 

11.49 

.  0.62645 

62 

8.55 

0.56541 

73 

5.65 

0.56205 

84 

2.875 

0.48390 

By  this  table  it  would  appear,  that,  as  turbines  are  generally  used,  a  velocity 
of  the  interior  circumference  of  the  wheel,  of  about  fifty-six  per  cent,  of  that  due 
to  the  fall  acting  upon  the  wheel,  would  be  most  suitable.  By  reference  to  the 
diagram  at  plate  VI.,  it  will  be  seen  that,  at  this  velocity  when  the  gate  is  fully 
raised,  the  coefficient  of  effect  will  be  within  less  than  one  per  cent,  of  the 
maximum. 

Other  considerations,  however,  must  usually  be  taken  into  account,  in  deter- 
mining the  velocity ;  the  most  frequent  is  the  variation  of  the  fall  under  which 
the  wheel  is  intended  to  operate.  If,  for  instance,  it  was  required  to  establish  a 
turbine  of  a  given  power,  on  a  fall  liable  to  be  diminished  to  one  half,  by 
backwater,  and,  that  the  turbine  should,  be  of  a  capacity  to  give  the  requisite 
power  at  all  times;  in  this  case,  the  dimensions  of  the  turbine  must  be  deter- 
mined for  the  smallest  fall;  but  if  it  has  assigned  to  it  a  velocity,  to  give  the 
maximum  effect  at  the  smallest  fall,  it  will  evidently  move  too  slow  for  the 
greatest  fall;  and  this  is  the  more  objectionable,  as,  usually,  when  the  fall  is 
greatest,  the  quantity  of  water  is  the  least,  and  it  is  of  the  most  importance  to 
obtain  a  good  effect.  It  would  then  be  usually,  the  best  arrangement,  to  give 
the  wheel  a  velocity  corresponding  to  the  maximum  coefficient  of  effect,  when 
the  fall  is  the  greatest.  To  assign  this  velocity,  we  must  first  find  the  propor- 
tional height  of  gate,  when  the  fall  is  greatest;  this  may  be  determined  approxi- 
mately by  aid  of  the  experiments  on  the  Tremont  Turbine. 

We  have  seen  that  P  =  0.085  Qh. 

Now,  if  h  is  increased  to  2h,  the  velocity,  and,  consequently,  the  quantity  of 
water  discharged,  will  be  increased  in  the  proportion  of  ^h  to  y/2A;  that  is  to 
say,  the  quantity  for  the  fall  2  h,  will  be  ~ 


RULES   FOR  PROPORTIONING  TURBINES.  49 

V 

Calling  P1  the  total  power  of  the  turbine  on  the  double  fall,  we  have 


F=Q.Q85  \f2Q2h, 
or  P'  =  0.085  X  2.8284  Qh. 

Thus,  the  total  power  of  the  turbine  is  increased  2.8284  times,  by  doubling 
the  fall  ;  on  the  double  fall,  therefore,  in  order  to  preserve  the  effective  power 
uniform,  the  regulating  gate  must  be  shut  down  to  a  point  that  will  give  only 
S.'OST  Part  °f  the  total  power  of  the  turbine. 

In  experiment  15,  the  fall  acting  upon  the  wheel  was  12.888  feet,  and  the 
total  useful  effect  of  the  turbine  was  85625.3  pounds  raised  one  foot  per  second  ; 
s-'ST'ST  Part  °f  this  *s  30273.4  Ibs.  ;  consequently,  the  same  opening  of  gate  that 
would  give  this  last  power,  on  a  fall  of  12.888  feet,  would  give  a  power  of 
85625.3  Ibs.  raised  one  foot  per  second,  on  a  fall  of  2  X  12.888  feet  =  25.776 
feet  To  find  this  opening  of  gate,  we  must  have  recourse  to  some  of  the 
other  experiments. 

In  experiment  73,  the  fall  was  13.310  feet,  the  height  of  gate  5.65  inches,  and 
the  useful  effect  58830.1  pounds.  In  experiment  83,  the  fall  was  13.435  feet,  the 
height  of  gate  2.875  inches,  and  the  useful  effect,  27310.9  pounds.  Reducing  both 
these  useful  effects  to  what  they  would  have  been,  if  the  fall  was  12.888  feet,  — 


the  useful  effect  in  experiment  73,     58830.1  (j||f§)  =  56054.5, 

3 

«  "  «  83,    27310.9  (JJ^I)  =  25660.1. 


By  a  comparison  of  these  useful  effects  with  the  corresponding  heights  of 
gate,  we  find,  by  simple  proportion  of  the  differences,  that  a  useful  effect  £>f 
30273.4  pounds  raised  one  foot  high  per  second,  would  be  given  when  the 
height  of  the  regulating  gate  was  3.296  inches. 

By   another  mode  :  — 

as  25660.1  :  2.875  ::  30273.4  :  2.875  X  8°2^'4  =  3.392  inches, 

2obo0.1 


a  little  consideration  will  show,  that  the  first  mode  must  give  too  little,  and  the 
second,  too  much;  taking  a  mean  of  the  two  results,  we  have  for  the  height 
of  the  gate,  giving  ^-.g-J^  of  the  total  power  of  the  turbine,  3.344  inches. 
Referring  to  table  II.,  we  see  that,  with  this  height  of  gate,  in  order  to  obtain 
the  best  coefficient  of  useful  effect,  the  velocity  of  the  interior  circumference  of 

7 


50  RULES  FOR  PROPORTIONING  TURBINES. 

the  wheel,  should  be  about  one  half  of  that  due  to  the  fall  acting  upon  the 
wheel ;  and  by  comparison  of  experiments  74  and  84,  it  will  be  seen  that,  with 
this  height  of  gate,  and  with  this  velocity,  the  coefficient  of  useful  effect  must 
be  near  0.50. 

This  example  shows,  in  a  strong  light,  the  well-known  defect  of  the  turbine, 
viz.,  giving  a  diminished  coefficient  of  useful  effect,  at  times  when  it  is  important 
to  obtain  the  best  results.  One  remedy  for  this  defect  would  be,  to  have  a 
spare  turbine,  to  be  used  when  the  fall  is  greatly  diminished ;  this  arrange- 
ment would  permit  the  principal  turbine  to  be  made  nearly  of  the  dimensions 
required  for  the  greatest  fall.  As  at  other  heights  of  the  water,  economy  of 
water  is  usually  of  less  importance,  the  spare  turbine  might  generally  be  of  a 
cheaper  construction. 

95.  To  lay  out  the  curve  of  the  buckets,  the  author  makes  use  of  the  following 
method. 

Keferring  to  plate  III.,  figure  1,  the  number  of  buckets,  N,  having  been  deter- 
mined by  the  preceding  rules,  set  off  the  arc  gi=^ — 

Let  (a=ffh,  the  shortest  distance  between  the  buckets; 
t  =  the  thickness  of  the  metal  forming  the  buckets. 

Make  the  arc  gjc=.5(a.  Draw  the  radius  Ok,  intersecting  the  interior  cir- 
cumference of  the  wheel  at  I;  the  point  I  will  be  the  inner  extremity  of  the 
bucket.  Draw  the  directrix  Im  tangent  to  the  inner  circumference  of  the  wheel. 
Draw  the  arc  on,  with  the  radius  w-f-tf,  from  i,  as  a  centre;  the  other  directrix, 
gp,  must  be  found  by  trial,  the  required  conditions  being,  that,  when  the  line 
ml  is  revolved  round  to  the  position  gt,  the  point  m  being  constantly  on  the 
directrix  gp,  and  another  point  at  the  distance  mg  =  rs,  from  the  extremity  of 
the  line  describing  the  bucket,  being  constantly  on  the  directrix  ml,  the  curve 
described  shall  just  touch  the  arc  no.  A  convenient  line  for  a  first  approxima- 
tion, may  be  drawn  by  making  the  angle  Ogp  =  11°.  After  determining  the 
directrix  according  to  the  preceding  method,  if  the  angle  Ogp  should  be  greater 
than  ]2°,  or  less  than  10°,  the  length  of  the  arc  g  k  should  be  changed,  to  bring 
the  angle  within  these  limits. 

The  curve  gss's"l,  described  as  above,  is  nearly  the  quarter  of  an  ellipse, 
and  would  be  precisely  so,  if  the  angle  gml  was  a  right  angle;  the  curve  may 
be  readily  described,  mechanically,  with  an  apparatus  similar  to  the  elliptic  tram- 
mel ;  there  is,  however,  no  difficulty  in  drawing  it  by  a  series  of  points,  as  is 
sufficiently  obvious. 


RULES   FOR   PROPORTIONING   TURBINES.  51 

96.  The    trace    adopted    by    the    author,    for   the    corresponding    guides,    is   as 
follows. 

The    number    n    having    been    determined,    divide    the    circle,    in    which    the 
extremities  of  the   guides   are   found,  into   n   equal   parts,  vw,  wx,  etc. 
Put  o/  for  the  width  between  two  adjoining  guides, 
and  t  for  the  thickness  of  the  metal  forming  the  guides. 
We  have  by  rule  4,  a/  =  —  . 

With  w  as  a  centre,  and  the  radius  (a'-\-if,  draw  the  arc  yz;  and  with  #  as  a 
centre,  and  the  radius  2  (a/  -(-  t'],  draw  the  arc  a'  b'.  Through  v  draw  the  portion 
of  a  circle  v  c',  touching  the  arcs  y  z  and  a'  V  ;  this  will  be  the  curve  for  the 
essential  part  of  the  guide.  The  remainder  of  the  guide,  c'd',  should  be  drawn 
tangent  to  the  curve  c'v  ;  a  convenient  radius  is  one  that  would  cause  the  curve 
c'd',  if  continued,  to  pass  through  the  centre  0.  This  part  of  the  guide  might 
be  dispensed  with,  except  that  it  affords  great  support  to  the  part  c'v,  and  thus 
permits  the  use  of  much  thinner  iron  than  would  be  necessary,  if  the  guide  ter- 
minated at  c',  or  near  it. 

97.  Collecting    together    the    foregoing    formulas    for    proportioning    turbines, 
which,  it  is  understood,   are   to   be  limited  to  falls  not  exceeding    forty   feet,   and 
to   diameters  not  less  than  two   feet;    we   have 

for  the  horse-power, 

P=  0.0425  2?  h  \JJ', 
for  the  diameter, 


for  the  quantity  of  water  discharged  per  second, 


for  the  velocity  of  the  interior  circumference  of  the  wheel,  when  the  fall  is  not 
very  variable, 

v  =  0.56  \l2gh, 
or,  v  —  4.491VJ; 

•  x 

for  the  height  of  the  orifices  of  discharge, 


52  RULES   FOR  PROPORTIONING   TURBINES. 

for  the  number  of  buckets, 


for  the  shortest  distance  between  two  adjacent  buckets, 

-=£> 

for  the  width  of  the  crown  occupied  by  the  buckets, 

W—  *D- 

"W 

for  the   interior   diameter   of  the   wheel, 

,7       n       8-° 
d  =  D-  —  ; 

for  the   number   of  guides, 

»  =  0.50^  to  0.75  N; 

for  the   shortest   distance   between  two   adjacent   guides, 


Table   IV.  has  been   computed   by   these   formulas. 

For  falls  greater  than  forty  feet,  the  height  of  the  orifices  in  the  circum- 
ference of  the  wheel,  should  be  diminished ;  the  foregoing  formulas  may,  however, 
still  be  made  use  of;  thus,  supposing  that  for  a  high  fall,  it  is  determined  to  make 
the  orifices  three  fourths  of  that  given  by  the  formula ;  divide  the  given  power,  or 
quantity  of  water  to  be  used,  by  0.75,  and  use  the  quotient  in  place  of  the 
true  power,  or  quantity,  in  determining  the  dimensions  of  the  turbine ;  no  modi- 
fication of  the  dimensions  will  be  necessary,  except  that  -fa  of  the  diameter  of 
the  turbine  should  be  diminished  to  -fa  of  the  diameter,  to  give  the  height  of 
the  orifices  in  the  circumference. 

98.  It  is  plain,  from  the  method  by  which  the  preceding  formulas  have 
been  obtained,  that  they  cannot  be  considered  as  established,  but  should  only  be 
taken  as  guides  in  practical  applications,  until  some  more  satisfactory  are  pro- 
posed, or  the  intricacies  of  the  turbine  have  been  more  fully  unravelled.  The 
turbine  has  been  an  object  of  deep  interest  to  many  learned  mathematicians, 
but,  up  to  this  time,  the  results  of  their  investigations,  so  far  as  they  have  been 
published,  have  afforded  but  little  aid  to  Hydraulic  Engineers. 


RULES   FOR   PROPORTIONING  TURBINES. 


53 


TABLE    IV. 

Table  for  Turbines  of  different  diameters,  operating  on  different  falls;  assuming  that  the  useful  effect  is 
seventy-Jive  per  cent,  of  the  power  expended ;  also  that  the  velocity  of  the  interior  circumference  is 
fifty-six  per  cent,  of  the  velocity  due  the  fall;  and  also  that  the  height  between  the  crowns  is  fa 
of  the  outside  diameter.  ' 


Outside  diameter  2.000  feet. 

Outside  diameter  3.000  feet. 

Outside  diameter  4.000  feet. 

Outside  diameter  5.000  feet. 

Outside  diameter  6.000  feet. 

Inside        "          1.656    " 

Inside        "          2.385    " 

Inside        "          3.238  " 

Inside        "           4.111    " 

Inside        "          5.000  " 

Fall 

Number  of  backets  36. 

Number  of  buckets  89. 

Number  of  buckets  42, 

Number  of  buckets  45. 

Number  of  buckets  48. 

in 

Quantity 

Quantity 

Quantity 

Quantity 

of  water 

Number 

of  water 

Number 

of  water 

Number 

of  water 

Number 

Quantity 

Number 

feet. 

dis- 

Number 

of 

dis- 

Number 

of 

dis- 

Number 

of 

dis- 

Number 

of 

of  water 

Number 

of 

charged 

of 

revolu- 

charged 

of 

revolu- 

charged 

of  horse- 

revolu- 

charged 

of  horse- 

revolu- 

discharged 

of  horse- 

revolu- 

in cubic 

horse- 

tions 

in  cubic 

horse- 

tions 

in  cubic 

power. 

tions 

in  cubic 

power. 

tions 

in  cubic 

power. 

tions 

feet  per 

power. 

per 

feet  per 

power. 

per 

feet  per 

per 

feet  per 

per 

feet  per 

per 

second. 

minute. 

second. 

minute. 

second. 

minute. 

second. 

minute. 

second. 

minute. 

5 

4.47 

1.90 

123.3 

10.06 

4.28 

80.4 

17.88 

7.60 

59.2 

27.95 

11.88 

46.7 

40.25 

17.11 

38.4 

6 

4.90 

2.50 

135.1 

11.02 

5.62 

88.1 

19.60 

9.99 

64.9 

30.62 

15.61 

51.1 

44.09 

22.49 

42.0 

7 

5.29 

3.15 

145.9 

11.91 

7.08 

95.2 

21.17 

12.59 

70.1 

33.07 

19.68 

55.2 

47.62 

28.34 

45.4 

8 

5.66 

3.85 

156.0 

12.73 

8.66 

101.7 

22.63 

15.39 

74.9 

35.35 

24.04 

59.0 

50.91 

34.62 

48.5 

9 

6.00 

4.59 

165.4 

13.50 

10.33 

107.9 

24.00 

18.36 

79.5 

37.50 

28.69 

62.6 

54.00 

41.31 

51.5 

10 

6.32 

5.38 

174.4 

14.23 

12.10 

113.7 

25.30 

21.50 

83.8 

39.53 

33.60 

66.0 

56.92 

48.38 

54.2 

11 

6.63 

6.20 

182.9 

14.92 

13.95 

119.3 

26.53 

24.81 

87.9 

41.46 

38.76 

69.2 

59.70 

55.82 

56.9 

12 

6.93 

7.07 

191.0 

15.59 

15.90 

124.6 

27.71 

28.27 

91.8 

43.30 

44.17 

72.3 

62.36 

63.60 

59.4 

13 

7.21 

7.97 

198.8 

16.23 

17.93 

129.7 

28.84 

31.87 

95.5 

45.07 

49.80 

75.2 

64.90 

71.72 

61.9 

14 

7.48 

8.90 

206.3 

16.84 

20.04 

134.6 

29.93 

35.62 

99.1 

46.77 

55.66 

78.1 

67.35 

80.15 

64.2 

15 

7.75 

9.88 

213.5 

17.43 

22.22 

139.3 

30.98 

39.50 

102.6 

48.41 

61.72 

80.8 

69.71 

88.88 

66.4 

16 

8.00 

10.88 

220.5 

18.00 

24.48 

143.9 

32.00 

43.52 

106.0 

50.00 

68.00 

83.5 

72.00 

97.92 

68.6 

17 

8.25 

11.92 

227.3 

18.55 

26.80 

148.3 

32.99 

47.66 

109.2 

51.54 

74.47 

86.0 

74.22 

107.24 

70.7 

18 

8.49 

12.98 

233.9 

19.09 

29.21 

152.6 

33.94 

51.93 

112.4 

53.03 

81.14 

88.5 

76.37 

116.84 

72.8 

19 

8.72 

14.08 

240.3 

19.61 

31.68 

156.8 

34.87 

56.32 

115.5 

54.49 

87.99 

90.9 

78.46 

126.71 

74.8 

20 

8.94 

15.21 

246.6 

20.12 

34.21 

160.9 

35.78 

60.82 

118.5 

55.90 

95.03 

93.3 

80.50 

136.84 

76.7 

21 

9.17 

16.36 

252.7 

20.62 

36.81 

164.8 

36.66 

65.44 

121.4 

57.28 

102.25 

95.6 

82.49 

147.24 

78.6 

22 

9.38 

17.54 

258.6 

21.11 

39.47 

168.7 

37.52 

70.17 

124.2 

58.63 

109.64 

97.9 

84.43 

157.88 

80.5 

23 

9.59 

18.75 

264.4 

21.58 

42.19 

172.5 

38.37 

75.01 

127.0 

59.95 

117.20 

100.1 

86.32 

168.76 

82.3 

24 

9.80 

19.99 

270.1 

22.04 

44.97 

176.2 

39.19 

79.95 

129.8 

61.24 

124.92 

102.2 

88.18 

179.89 

84.0 

25 

10.00 

21.25 

275.7 

22.50 

47.81 

179.8 

40.00 

85.00 

132.4 

62.50 

132.81 

104.3 

90.00 

191.25 

85.8 

26 

10.20 

22.54 

281.1 

22.95 

50.71 

183.4 

40.79 

90.15 

135.1 

63.74 

140.86 

106.4 

91.78 

202.84 

87.5 

27 

10.39 

23.85 

286.5 

23.38 

53.66 

186.9 

41.57 

95.40 

137.6 

64.95 

149.06 

108.4 

93.53 

214.65 

89.1 

28 

10.58 

25.19 

291.8 

23.81 

56.67 

190.3 

42.33 

100.75 

140.2 

66.14 

157.42 

110.4 

95.25 

226.69 

90.8 

29 

10.77 

26.55 

296.9 

24.23 

59.73 

193.7 

43.08 

106.20 

142.6 

67.31 

165.93 

112.4 

96.93 

238.94 

92.4 

30 

10.95 

27.93 

302.0 

24.65 

62.85 

197.0 

43.82 

111.74 

145.1 

68.46 

174.59 

114.3 

98.59 

251.41 

94.0 

31 

11.14 

29.34 

307.0 

25.05 

66.02 

200.3 

44.54 

117.37 

147.5 

69.60 

183.39 

116.2 

100.22 

264.08 

95.5 

32 

11.31 

30.77 

311.9 

25.46 

69.24 

203.5 

45.25 

123.09 

149.8 

70.71 

192.33 

118.0 

101.82 

276.96 

97.0 

33 

11.49 

32.23 

316.7 

25.85 

72.51 

206.6 

45.96 

128.91 

152.2 

71.81 

201.42 

119.9 

103.40 

290.04 

98.5 

34 

11.66 

33.70 

321.5 

26.24 

75.83 

209.7 

46.65 

134.81 

154.5 

72.89 

210.64 

121.7 

104.96 

303.33 

100.0 

35 

11.83 

35.20 

326.2 

26.62 

79.20 

212.8 

47.33 

140.80 

156.7 

73.95 

220.00 

123.4 

106.49 

316.81 

101.5 

36 

12.00 

36.72 

330.8 

27.00 

82.62 

215.8 

48.00 

146.88 

158.9 

75.00 

229.50 

125.2 

108.00 

330.48 

102.9 

37 

12.17 

38.26 

335.4 

27.37 

86.09 

218.8 

48.66 

153.04 

161.1 

76.03 

239.13 

126.9 

109.49 

344.34 

104.3 

38 

12.33 

39.82 

339.9 

27.74 

89.60 

221.7 

49.32 

159.29 

163.3 

77.05 

248.89 

128.6 

110.96 

358.40 

105.7 

39 

12.49 

41.40 

344.3 

28.10 

93.16 

224.6 

49.96 

165.62 

165.4 

78.06 

258.78 

130.3 

112.41 

372.64 

107.1 

40 

12.65 

43.01 

348.7 

28.46 

96.77 

227.5 

50.60 

172.03 

167.5 

79.06 

268.79 

132.0 

113.84 

387.06 

108.5 

54 


KULES  FOR  PROPORTIONING  TURBINES. 


TABLE    IV. —  CONTINUED. 


Outside  diameter  7.000  feet. 

Outside  diameter  8.000  feet. 

Outside  diameter  9.000  feet 

Outside  diameter  10.000  feet. 

Inside        "          5.902    " 

Inside        "          6.815    " 

Inside        "          7.737  " 

Inside         "          8.667  " 

Number  of  buckets  51. 

Number  of  buckets  54. 

Number  of  buckets  57. 

Number  of  buckets  60. 

Fall 
in 

Quantity  of 

Number 

Quantity  of 

Number 

Quantity  of 

Number 

Quantity  of 

Number 

feet 

water 

Number 

of 

water 

Number 

of 

water 

Number 

of 

water 

Number 

of 

discharged, 

of 

revolu- 

discharged, 

of 

revolu- 

discharged, 

of  horse- 

revolu- 

discharged 

of  horse- 

revolu- 

in cubic 

horse- 

tions 

in  cubic 

horse- 

tions 

in  cubic 

power. 

tions 

in  cubic 

power. 

tions 

feet  per 

power. 

per 

feet  per 

power. 

per 

feet  per 

per 

feet  per 

per 

second. 

minute. 

second. 

minute. 

second. 

minute. 

second. 

minute. 

5 

54.78 

23.28:     32.5 

71.55 

30.41 

28.1 

90.56 

38.49 

24.8 

111.80 

47.52 

22.1 

6 

60.01 

30.61 

35.6 

78.38 

39.97 

30.8 

"  99.20 

50.59 

27.2 

122.47 

62.46 

24.2 

7 

64.82 

38.57 

38.4 

84.67 

50.37 

33.3 

107.15 

63.76 

29.3 

132.29 

78.71 

26.2 

8 

69.30 

47.12 

41.1 

90.51 

61.55 

35.6 

114.55 

77.90 

31.4 

141.42 

96.17 

28.0 

9 

73.50 

56.23 

43.6 

96.00 

73.44 

37.8 

121.50 

92.95 

33.3 

150.00 

114.75 

29.7 

10 

77.47 

65.86 

46.0 

101.19 

86.02 

39.8 

128.07 

108.86 

35.1 

158.11 

134.40 

31.3 

11 

81.26 

75.97 

48.2 

106.13 

99.23 

41.7 

134.32 

125.59 

36.8 

165.83 

155.05 

32.8 

12 

84.87 

86.57 

50.3 

110.85 

113.07 

43.6 

140.30 

143.10 

38.4 

173.21 

176.67 

34.3 

13 

88.34 

97.61 

52.4 

115.38 

127.49 

45.4 

146.03 

161.36 

40.0 

180.28 

199.21 

35.7 

14 

91.67 

109.09 

54.4 

119.73 

142.48 

47.1 

151.53 

180.33 

41.5 

187.08 

222.63 

37.0 

15 

94.89 

120.98 

56.3 

123.94 

158.02 

48.7 

156.86 

199.99 

42.9 

193.65 

246;90 

38.3 

16 

98.00 

133.28 

58.1 

128.00 

174.08 

50.3 

162.00 

220.32 

44.3 

200.00 

272.00 

39.6 

17 

101.02 

145.97 

59.9 

131.94 

190.65 

51.9 

166.99 

241.29 

45.7 

206.16 

297.89 

40.8 

18 

103.94 

159.03 

61.7 

135.76 

207.72 

53.4 

171.83 

262.89 

47.0 

212.13 

324.56 

42.0 

19 

106.79 

172.47 

63.3 

139.48 

225.27 

54.9 

176.53 

285.10 

48.3 

217.94 

351.98 

43.1 

20 

109.57 

186.26 

65.0 

143.11 

243.28 

56.3 

181.12 

307.91 

49.6 

223.61 

380.13 

44.3 

21 

112.27 

200.41 

66.6 

146.64 

261.75 

57.7 

185.60 

331.28 

50.8 

229.13 

408.99 

45.4 

22 

114.91 

214.89 

68.2 

150.09 

280.67 

59.0 

189.96 

355.23 

52.0 

234.52 

438.55 

46.4 

23 

117.50 

229.71 

69.7 

153.47 

300.03 

60.4 

194.23 

379.72 

53.2 

239.79 

468.79 

47.5 

24 

120.02 

244.85 

71.2 

156.77 

319.81 

61.7 

198.41 

404.76 

54.3 

244.95 

499.70 

48.5 

25 

122.50 

260.31 

72.7 

160.00 

340.00 

62.9 

202.50 

430.31 

55.4 

250.00 

531.25 

49.5 

26 

124.93 

276.09 

74.1 

163.17 

360.60 

64.2 

206.51 

456.39 

56.5 

254.95 

563.44 

50.5 

27 

127.30 

292.17 

75.5 

166.28 

381.61 

65.4 

210.45 

482.97 

57.6 

259.81 

596.26 

51.4 

28 

129.64 

308.55 

76.9 

169.33 

403.00 

66.6 

214.31 

510.05 

58.7 

264.58 

629.69 

52.4 

29 

131.93 

325.22 

78.3 

172.32 

424.78 

67.8 

218.09 

537.61 

59.7 

269.26 

663.72 

53.3 

30 

134.19 

342.19 

79.6 

175.27 

446.94 

68.9 

221.83 

565.66 

60.7 

273.86 

698.35 

54.2 

31 

136.41 

359.44 

80.9 

178.17 

469.47 

70.1 

225.50 

594.18 

61.7 

278.39 

733.55 

55.1 

32 

138.59 

376.97 

82.2 

181.02 

492.37 

71.2 

229.10 

623.16 

62.7 

282.84 

769.33 

56.0 

33 

140.74 

394.78 

83.5 

183.82 

515.63 

72.3 

232.66 

652.59 

63.7 

287.23 

805.67 

56.9 

34 

142.86 

412.86 

84.7 

186.59 

539.24 

73.4 

236.16 

682.48 

64.6 

291.55 

842.57 

57.7 

35 

144.94 

431.21 

86.0 

189.31 

563.21 

74.5 

239.60 

712.82 

65.6 

295.80 

880.02 

58.5 

36 

147.00 

449.82 

87.2 

192.00 

587.52 

75.5 

243.00 

743.58 

66.5 

300.00 

918.00 

59.4 

37 

149.03      468.69 

88.4 

194.65 

612.17 

76.6 

246.35 

774.77 

67.4 

304.14 

956.51 

60.2 

38 

151.03 

487.82 

89.6 

197.26 

637.15 

77.6 

249.66 

806.40 

68.3 

308.22 

995.55 

61.0 

39 

153.00 

507.20 

90.8 

199.84 

662.47 

78.6 

252.92 

838.44 

69.2 

312.25 

1035.11 

61.8 

40 

164.95 

526.83 

91.9 

202.39 

688.12 

79.6 

256.15 

870.89 

70.1 

316.23 

1075.17 

62.6 

55 


EXPERIMENTS  ON  A  MODEL  OF  A  CENTJIE-VENT  WATER-WHEEL,   WITH 

STRAIGHT  BUCKETS. 


99.  THE  author  was  led  to   this   design  by  the   consideration   of  the   path   of 
the   water   in   passing   through   the   wheel,   according    to    the    hypothesis  in   art.  83. 
It  is  a  wheel  well   suited   for  low   falls,  in   which  the   water,  over  the   wheel,  may 
stand    at    its    natural    height,    without    requiring    a   vertical   shaft   of    great   length. 
Its   simplicity  and   cheapness,  combined  with   its   other   good    qualities  as  a  hydraulic 
motor,  must   recommend   it   for   many   such   situations. 

100.  Plate   VII.,  figure   1,   is  a   general  plan,   and  figure   2,  a  vertical  section 
of  the   apparatus. 

Figure  3  is  a  vertical  section  through  the  apertures  in  the  guides  and  wheel; 
the  guides  and  buckets  are  omitted  to  avoid  confusion  in  the  figure. 

Figure  4  is  a  horizontal  section  of  part  of  the  guides  and  buckets,  showing, 
also,  the  path  of  the  water  in  experiment  3,  according  to  the  hypothesis  in  art.  83. 

A  is  the  wheel ;  the  exterior  diameter  is  22|-  inches ;  the  interior  diameter 
is  19£  inches;  the  height  between  the  crowns,  or  B  C,  figure  3,  is  2||  inches; 
it  carries  thirty-six  buckets,  EE,  figure  4,  of  steel,  about  -fa  of  an  inch  in  thick- 
ness, fastened  to  the  wheel  by  means  of  the  wooden  cushions  F  F,  figure  3 ;  the 
upper  cushions  are  screwed  to  the  disc  D,  and  the  lower  ones  to  the  crown  G.  The 
disc  D  is  of  castriron,  f  inch  thick,  with  a  suitable  hub  by  which  it  is  connected 
with  the  vertical  shaft. 

HH  are  guides  of  cast-iron,  which  direct  the  water  into  the  wheel,  and  also 
support  the  plate  I,  which  protects  the  wheel  from  pressure  on  its  upper  surface ; 
the  contraction  of  the  streams  entering  the  apertures  between  the  guides,  is  dimin- 
ished by  the  curved  wooden  garniture  K;  there  are  twenty-four  guides.  The  mean 
shortest  distance  between  the  buckets  at  ab,  figure  4,  is  0.0339  feet;  the  mean 
shortest  distance  between  the  guides  cd,  figure  4,  is  0.0437  feet;  and  the  height 
of  both  is  2|  |  inches  —  0.2344  feet ;  we  have,  therefore,  for  the  sum  of  the 
areas  of  the  smallest  sections  between  the  guides, 

0.0437  X  0.2344  X  24  =  0.24584  square  feet. 


56  EXPERIMENTS   ON   A  MODEL   OF   A 

Similarly,   the   sum   of   the   areas   of   the   smallest  sections   between   the   buckets  is 
0.0339  X  0.2344  X  36  =  0.28606  square   feet. 

| 

The  water  is  admitted  into  the  forebay  L,  by  the  pipes  MM;  the  diaphragm 
N  is  to  diminish  the  agitation  of  the  water. 

101.  The   apparatus  for    gauging    the   water    discharged    by    the    wheel,   con- 
sisted   of    the    weir     0,    which    had    sharp    edges;     the    depth    on    the    weir   was 
measured    by   a    hook    gauge,    in    the    box    P,   which    communicated,   by   a    small 
aperture,  with   the    surrounding   water;    the   height   of  the    water   above    the   wheel 
was   taken   at   a   gauge   in   the   box    Q;    this    box  was  made   sloping   on   one    side, 
in   order  to   permit   a   better  view   of  the    gauge.     The    zeros   of  both  gauges  were 
at   the    level   of    the   top    of    the    weir;     consequently,   the   difference   in   the   read- 
ings of  the   gauges   gave   at   once   the   fall   acting  upon   the   wheel. 

102.  The  apparatus  for  measuring  the  power,  consisted  of  the  Prony  dynamom- 
eter R,  attached  to  the  upper  part  of  the  vertical  shaft ;    the  weights  were  applied 
by   means   of  the   bell   crank    S,  figures   1,  2,  and    5  ;    the  oscillations  of  the  brake 
were   diminished   by   the   hydraulic   regulator   T,  and   the    extent   of  the   oscillations 
was  limited   by  the   stops   UU.      The   speed  of  the   wheel  was   obtained   by  means 
of    a   counter,   driven   by   the   worm    V,   attached   to   the   top   of  the   upright  shaft; 
this  was  so   arranged   as   to   strike   a  bell   once   in  fifty  revolutions  of  the  wheel. 

In  order  to  diminish  the  passive  resistances,  the  weight,  bearing  upon  the 
step  W,  was  counterbalanced,  in  part,  by  other  weights,  one  of  which  is  represented 
at  y,  figure  2 ;  these  were  attached  to  the  brakes  at  the  points  2&X,  by  vertical 
cords  passing  over  pulleys ;  the  weight,  resting  on  the  step  when  the  wheel  was 
immersed,  and  the  dynamometer  attached,  was  found  to  be  170  pounds;  the  coun- 
terbalance was  160  pounds,  leaving  10  pounds  bearing  upon  the  step.  The 
entire  apparatus  for  measuring  the  power,  was  in  equilibrium  when  there  were 
no  weights  in  the  scale. 

103.  In  all   the    experiments,  except   experiment  10,  the  brake  was   lubricated 
with   oil ;    in   experiment    10   water  was   used  for  this  purpose ;    experiments  9   and 
10   were   identical   in   all   other   respects.      It   was   noticed    in    experiment    10    that 
the  whole   apparatus  trembled  very  much ;    this  must   have   consumed    some  power, 
which   is   perceptible  in  the  coefficients  of  effect.      Experiment    9,  in  which  oil  was 
used,  and   in  which  the   trembling   of  the    apparatus  was  very  slight,  gives   a  coeffi- 
cient   of    effect    of    0.6922;     while    experiment    10,   in   which    water   was    used    to 
lubricate   the   brake,   and   in   which   the   trembling   of    the   apparatus   was   very    dis- 
tinct, gave   0.6886   as   the   coefficient   of  effect. 


CENTRE-VENT   WATER-WHEEL,  WITH   STRAIGHT   BUCKETS.  57 

\ 

104.  All   the   apparatus  was   constructed   with   great   care   and    precision;    the 
surfaces    of   the    cast-iron   guides   were   ground   smooth ;    and    the    cast-iron    disc    and 
lower   crown   of   the   wheel   were   turned   true,   and   polished,   in   order   to   diminish, 
as  much   as  possible,   the   resistance   of  the   water   to   the   motion   of  the   wheel. 

105.  In   table   V.,   the   quantity   of    water   discharged   has  been   calculated   by 
the   formula 

#  =  3.33(7—  Q.lnh)$, 

in  which  Q  =  ihe  quantity  in  cubic  feet  per  second;  l  =  the  length  of  the 
weir  =  3.003  feet ;  n  =  the  number  of  end  contractions  —  2  ;  h  =  the  depth  upon 
the  weir.  The  weights  were  obtained  for  the  purpose  from  Mr.  0.  A.  Richard- 
son, the  official  sealer  of  weights  and  measures  for  the  City  of  Lowell.  The 
effective  length  of  the  lever  of  the  dynamometer,  was  two  feet.  The  tempera- 
ture of  the  water  was  63£°  Fahrenheit.  Temperature  of  the  air  at  8h,  35'  A.  M., 
63°  Fahrenheit.  The  weight  of  a  cubic  foot  of  water  is  taken  at  62.3128  pounds, 
which  is  deduced  from  table  I. 

If,  in  any  experiment,  the  brake  touched,  even  momentarily,  either  of  the 
stops  UU,  it  was  rejected;  with  the  use,  however,  of  a  regular  and  sufficient 
quantity  of  oil  to  lubricate  the  brake,  and  a  properly  constructed  hydraulic  reg- 
ulator, there  is  seldom  any  difficulty  from  this  cause,  except  at  very  low  velocities. 

8 


58 


EXPERIMENTS   ON  A  MODEL   OF   A 


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CENTRE-VENT   WATER-WHEEL,   WITH   STRAIGHT   BUCKETS. 


69 


106.  In  the  foregoing  table,  experiments  4,  5,  6,  and  7,  were  made  with  the 
•wheel  still ;  the  brake  was  screwed  up  tight,  and  the  pressure  of  the  water  upon 
the  buckets,  was  measured  by  weights  in  the  scale.  In  experiments  4  and  7, 
the  weights  were  sufficient  to  balance  the  effect  of  the  pressure  of  the  water  on 
the  buckets,  and  also  to  overcome  the  friction  of  the  apparatus;  in  other  words, 
the  weights  were  the  least  that  would  cause  the  scale  to  preponderate  over  the 
active  and  passive  forces.  In  experiments  5  and  6,  the  weights  in  the  scale  were 
the  greatest  that  the  pressure  upon  the  buckets  would  raise,  and  overcome  the 
friction  of  the  apparatus ;  consequently,  the  force  of  the  water  acting  upon  the 
buckets,  may  be  considered  as  balanced  by  the  average  of  the  weights  in  the 
fourth  and  fifth  experiments,  and,  also,  by  the  average  in  the  sixth  and  seventh 
experiments. 

To  obtain  the  true  weight  that  would  balance  the  pressure,  we  must  reduce 
the  weights  in  the  different  experiments  to  what  they  would  have  been,  if  the 
fall  acting  upon  the  wheel  had  been  constant. 

The  following  table  shows  the  weights  reduced  to  a  uniform  fall  of  2.5  feet, 
obtained  by  simple  proportion;  thus,  in  the  fourth  experiment, 

2.5160  :  25.75  ::  2.500  :  25.586. 
The   quantities   discharged   are   also   given  for  a   uniform   fall   of  2.5  feet. 


Number  of 
experiment. 

Actual  full  acting  upon 
the  wheel,  in  feet. 

Weight  in  scale  by 
experiment,  in  pounds. 

Weight  reduced  to  a 
uniform  fall  of  2.5  feet 

Quantity  of  water  dis- 
charged, reduced  to 
a  uniform  fall  of  2.5 
feet,  in  cubic   feet 
per  second. 

4 

2.5160 

25.750 

25.586 

1.9651 

5 

2.5074 

19.375 

19.318 

1.9567 

6 

2.4405 

19.250 

19.719 

1.9568 

7 

2.3735 

24.125 

25.411 

1.9584 

Means 

22.5085 

1.9592 

The   mean   reduced  weight,  when  the  weights  preponderated,  is  25.4985  pounds. 

and  when  the  pressure  on  the  buckets  preponderated,      .      19.5185       " 
Difference, 5.9800  pounds. 

Half  of  this  difference,  or  2.99  pounds,  may  be  considered  as  the  measure  of 
the  passive  resistances,  or,  rather,  of  the  friction  of  the  apparatus. 

107.  In  experiment  13,  the  brake  was  entirely  removed,  and  the  wheel 
allowed  to  run  without  load ;  with  the  brake,  the  counterbalance  was  necessarily 


60 


EXPERIMENTS  ON  A  MODEL  OF  A  CENTRE-VENT  WATER-WHEEL. 


removed,  consequently  the   passive   resistance   arising  from   the   friction  of  the   step, 
was   much  greater   than   hi   the   other   experiments. 

108.  Fig.   6,  plate    VII,  is    a    diagram    representing    the    experiments;     the 
abscissas  represent    the    ratios    of   the  velocities  of    the    exterior    circumference   of 
the  wheel,  to   the   velocities    due    to    the    falls    acting    upon    the   wheel,  as   given 
in    column    14,  of   table    V. ;     the    ordinates    represent    the    ratios    of    the   useful 
effects   to    the    powers  expended,  as    given    in   column   11 ;     the    points,   represent- 
ing experiments    12   and    13,   are   connected    by  a  broken  line,  because   the   latter 
experiment    is    not    strictly    comparable    with    the    others,    in    consequence    of    the 
removal   of  the   counterbalance. 

109.  The   following  table   contains  the   successive   steps   of  the   calculation   for 
the   ordinates  of  the   path   of  the   water  in   experiment   3,  represented   at   figure  4, 
plate   VII. ;    the   operations   are   all   similar   to   those   explained    in   articles   83   and 
119.     The   ordinates  in   column  10   are   obtained /by  the   formula 


0  = 


in   which 


0   is   the    ordinate, 

R  the   corresponding  value   of  the   radius  in   column   1, 


to,  the  angular  velocity  = 


850  X  2 
551.5 


;  9.684, 


AH,  the   corresponding   volume   in   column    9, 
Q",  the  mean  quantity  discharged  by  each  aperture  in  the  wheel: 


2.1681 
:    36 


=  0.06022. 


1 

a 

3 

4 

5 

6 

7 

8 

9 

10 

Value  of 
Ji  and 
successive 
values  of 
JZ',in 
inches. 

Areas  in 
square 
inches,  of 
circles  of 
the  radii  hi 
column  1. 

Areas  in 
square 
inches,  of 
the 
complete 
rings. 

Tfe 

of  the  areas 
of  the  rings 
in  column  3, 
in  square  feet. 

Correction  for 
the  thickness 
of  the  bucket, 
in  square  feet. 

True  areas 
of  the  partial 
rings,  in 
square  feet. 

Height 
of  the 
partial 
rings, 
infect. 

Volumes  of 
the  partial 
rings,  in 
cubic  feet. 

Volumes 
between  R 
and  the 
successive 
Talues  of  Rf, 
in  cubic  feet. 

Ordinates 
in  feet, 
measured 
on  arcp  of 
the  radii  in 
column  1. 

11.437 

410.936 

11.000 

380.133 

30.803 

0.005942 

0.000262 

0.005680 

0.2344 

0.001331 

0.001331 

0.1962 

10.500 

346.361 

33.772 

0.006515 

0.000350 

0.006165 

ft 

0.001445 

0.002776 

0.3906 

10.250 

330.064 

16.297 

0.003144 

0.000198 

0.002946 

a 

0.000691 

0.003467 

0.4762 

10.000 

314.159 

15.905 

0.003068 

0.000228 

0.002840 

u 

0.000666 

0.004133!  0.5538 

9.875 

306.354 

7.805 

0.001506 

0.000156 

0.001350 

tt 

0.000316 

0.004449 

0.5887 

9.750 

298.648 

7.706 

0.001486 

0.000175 

0.001311 

u 

0.000307 

0.004756 

0.6214 

61 


EXPERIMENTS  ON  THE  POWER  OF  A  CENTRE-VENT  WATER-WHEEL,   AT 
THE  BOOTT  COTTON-MILLS  IN  LOWELL,  MASSACHUSETTS. 


110.  THIS  wheel   is  one   of  a  pair  constructed  from  the  designs  of  the  author 
by   the    Lowell    Machine    Shop,    for    the    Boott    Cotton-Mills,   in   1849.      During  a 
considerable    portion  of  the   year,  the   fall,  on  which  these  wheels  operate,  is  about 
nineteen   feet ;    with   this   fall,   and   with    the    regulating    gates    raised    to   the   full 
height,  they   each   furnish   an   effective   power   of  about   230   horse-power. 

A  patent  for  the  term  of  fourteen  years  was  issued,  July  26,  1838,  by  the 
Government  of  the  United  States  of  America,  to  Samuel  B.  Howd,  of  Geneva,  in 
the  State  of  New  York,  for  a  water-wheel  resembling,  in  some  respects,  the  wheels 
at  the  Boott  Cotton-Mills.*  Under  this  patent,  a  large  number  of  wheels  have 
been  constructed,  and  a  great  many  of  them  are  now  running  in  different  parts 
of  the  country ;  they  are  known  in  some  places  as  the  Iloivd  wheel,  in  others  as 
the  United  States  wheel;  they  have  uniformly  been  constructed  in  a  very  simple 
and  cheap  manner,  in  order  to  meet  the  demands  of  a  numerous  class  of  millers 
and  manufacturers,  who  must  have  cheap  wheels  if  they  have  any. 

111.  Figures   3   and   4,  plate   IX.,  are   a   plan   and   vertical   section   of  one   of 
the    Howd   wheels,   constructed    by   the    owners   of    the   patent   right   for   a    portion 
of    New   England.      A,   the    wooden   guides   by   which    the    water   is   directed    on    to 
the   buckets ;    Z?,    buckets   of    castnron,    fastened    to    the    upper    and    lower   crowns 
of  the   wheel,  by  bolts ;    the   upper  crown   is   connected   with   the  vertical   shaft  E, 
by  the   arms  C.     D,  the   regulating   gate,  placed  outside  of  the  guides ;    this  is  made 
of  wood  ;    the    apparatus   by  which  it  is  moved  is  not  represented ;    it  is  a  simple 
arrangement   of   levers.      The   upright  shaft  E   runs    on    a    step    at    the    bottom. 
This   wheel   is    usually   placed    in   the    bottom    of   a   rectangular   forebay,   which,   in 
high   falls,   may   be   closed   at  the    top,  so   as   to    avoid    the    necessity   of   using  a 
vertical   shaft   of  great   length.     The   peculiarly  shaped   projections   on   one   side   of 
the   buckets,   it   is   said,   increase   the   efficiency   of    the   wheel,   by   diminishing   the 


*  A  wheel  similar,  in  its  essential  features,  was  proposed  in  France,  in  1826,  by  Poncekt. 


62  EXPERIMENTS   ON  A   CENTRE-VENT   WATER-WHEEL, 

waste  of  water;  it  is  possible  that  some  such  effect  may  be  produced  by  them. 
The  author  is  not  aware  that  any  exact  experiments  have  been  made  on  the 
power  of  these  wheels ;  from  their  form  and  construction,  however,  it  is  plain 
that  they  cannot  be  classed  among  those  using  water  with  very  -great  economy. 
In  the  design  for  the  Boott  wheel,  the  author  has  so  modified  the  form  and 
arrangement  of  the  whole,  as  to  produce  a  wheel  essentially  different  from  the 
Howd  wheel,  as  above  described,  although  it  may,  possibly,  be  technically  covered 
by  the  patent  for  that  wheel. 

112.  Figures  1  and  2,  plate  VIII.,  are  a  vertical  section,  and  a  plan  of  the 
Boott  centre-vent  wheel,  showing,  also,  the  apparatus  used  in  the  experiments.  A, 
the  lower  end  of  a  pipe,  about  one  hundred  and  thirty  feet  long,  and  eight  feet 
in  diameter,  by  which  the  water  is  conducted  into  the  forebay  B ;  this  pipe  is 
constructed  of  plate  iron,  three  eighths  of  an  inch  in  thickness,  riveted  together  in 
the  usual  manner  of  making  steam-boilers.  For  local  reasons,  the  top  of  the  fore- 
bay  B  is  closed,  so  as  to  prevent  the  water  from  rising  to  its  natural  level,  by 
about  six  or  seven  feet.  O,  the  surface  of  the  water  in  the  Merrimack  River, 
represented  at  about  its  medium  height  during  the  experiments.  Z>,  the  wheel; 
E,  the  guides ;  F,  the  regulating  gate,  the  apparatus  for  moving  which,  is  not  rep- 
resented ;  G,  the  disc,  which  relieves  the  wheel  from  the  vertical  pressure  of  the 
water,  and  which  also  supports  the  lower  bearing  of  the  vertical  shaft.  The 
leather  packing  of  the  regulating  gate  F,  slides  against  the  circumference  of  the 
disc,  which  is  turned  smooth  and  cylindrical  for  that  purpose,  and  the  disc  itself 
is  supported  by  means  of  four  brackets,  two  of  which  are  represented  at  HH, 
by  the  columns  II.  The  vertical  shaft  K  is  of  wrought  iron,  and  it  passes  through 
the  stuffing  box  L,  and  is  supported  by  the  box  M,  which  has  a  series  of  recesses 
lined  with  babbit  metal,  fitted  to  receive  a  corresponding  series  of  projections  in 
the  vertical  shaft.  The  wheel,  the  vertical  shaft,  and  the  bevel  gear  usually  on 
the  latter,  have  a  total  weight  of  about  15,200  pounds ;  the  bearing  surface  in 
the  box  M  is  about  331  square  inches,  consequently,  the  weight,  per  square  inch, 
of  bearing  surface,  is  about  46  pounds. 

Figures  3  and  4,  plate  VIII.,  represent  the  wheel  and  guides  on  a  larger 
scale.  The  buckets  and  guides  are  equal  in  number,  there  being  forty  of  each ; 
the  buckets  are  of  plate  iron,  ^  of  an  inch  in  thickness ;  the  guides  are  of  the 
same  material,  T8g  of  an  inch  in  thickness.  The  following  dimensions  were  taken 
after  the  parts  were  finished :  — 

Mean     shortest     distance     between     adjacent     buckets,     or     a  I 

figure    4, 0.1384  feet. 


AT   THE   BOOTT   COTTON-MILLS.  C3 

Mean   height   between   the   crowns,  at  the   inner    extremities  of 

the  'buckets,  or  cd,  figure  3, 1.2300  feet. 

Mean   height   between   the   crowns,  at  the   outer   extremities   of 

the   buckets,  or  ef,  figure  3, 0.9990  « 

Mean   shortest    distance    between    the    adjacent   guides,   or  gh, 

figure   4, 0.1467  " 

Mean  height  of  the  orifices  between  the  guides,  or  ik,  figure  3,     1.0066  " 

Diameter  of  the  wheel  at  the  outside  of  the  buckets,    ....     9.338  " 

Diameter  of  the  wheel  at  the  inside  of  the  buckets,      ....     7.987  " 

113.  Several  of  the  peculiar  features  of  this  design  are  covered  by  patents 
issued  by  the  Government  of  the  United  States  to  TJ.  A.  Boyden.  His  patents 
cover  the  arrangement  of  the  regulating  gate,  by  placing  it  between  the  guides 
and  the  wheel,  and  having  it  detached  from  the  garniture ;  making  the  height 
between  the  crowns  of  the  wheel  greater  where  the  water  is  discharged,  than 
where  it  enters;  they  also  cover  the  self-adjusting  apparatus  on  which  the  box 
M  is  supported. 

.  114.  Returning  to  figures  1  and  2,  -plate  VIII.,  N  is  the  friction  pulley  of 
the  dynamometer,  which  is  attached  to  the  part  of  the  shaft  intended  to  receive 
the  hub  of  the  bevel  gear,  for  the  transmission  of  the  power ;  0,  the  brake  of 
maple  wood ;  P,  the  bell  crank,  and  Q,  the  hydraulic  regulator ;  the  friction 
pulley  and  the  brake  were  subsequently  used  in  the  experiments  on  the  Tremont 
Turbine,  in  the  account  of  which  they  are  more  particularly  described,  (see  arts. 
37  and  38).  R,  the  weir  at  which  the  water  discharged  by  the  wheel  was 
gauged;  S,  a  grating  for  the  purpose  of  equalizing  the  flow  of  the  water  towards 
the  weir ;  T,  the  gauge  box  in  which  the  depths  on  the  weir  were  observed. 
The  communication  between  the  water  inside  the  box,  and  that  surrounding  it, 
was  maintained  by  means  of  an  aperture  in  the  bottom  of  the  box,  (which 
extended  1.06  feet  below  the  top  of  the  weir,)  and  which  was  4.12  feet  from  the 
weir.  It  may  be  thought,  at  first  sight,  that  the  depths  on  the  weir  were  taken 
so  near  it,  as  to  be  affected  by  the  curvature  in  the  surface,  caused  by  the 
discharge  over  the  weir,  but  the  experiments  at  the  Lower  Locks,  (art.  173,) 
prove,  conclusively,  that  when  the  communication  between  the  water  inside  the 
box,  and  that  outside  of  it,  is  maintained,  by  means  of  a  pipe  opening  near 
the  bottom  of  the  canal,  the  depths  are  not  affected  in  any  appreciable  degree, 
by  the  curvature  in  the  surface.  If  any  such  effect  was  produced  in  this  case, 
it  must  have  been  very  slight.  U  and  V  are  the  gauge  boxes  at  which  the 
heights  of  the  water,  below  and  above  the  wheel,  were  observed,  in  order  to 


64  EXPERIMENTS   ON  THE   BOOTT  CENTRE-VENT  WATER-WHEEL. 

obtain  the  fall  acting  upon  the  wheel.  The  velocity  of  the  wheel  was  obtained 
by  means  of  the  counter  W.  The  apparatus  for  lubricating  the  brake  is  not 
represented  on  the  plate  ;  in  some  of  the  experiments,  water  was  used,  and  in 
others,  linseed  oil. 

The  experiments  were  made  according  to  the  method  of  continuous  observa- 
tions, which  has  been  sufficiently  described  in  the  account  of  the  experiments  on 
the  Tremont  Turbine. 

115.  The    experiments    on    the    Boott    centre-  vent   water-wheel,   are   given   in 
detail    in    table   VI.,   which   will    be   intelligible,   without  much   further   explanation 
than   is   contained   in   the   respective   headings   of  the   several  columns. 

116.  COLUMN  10.      Useful  effect,   or  the  friction   of  the   brake,  in  pounds  avoirdupois 
raised  one  foot   per  second.      The    brake    was    connected    with   the    vertical  '  arm    of 
the    bell    crank,   by   a    link,    which    was    horizontal    when    the    brake   was    in    its 
normal   position.     When   in   this   position,  the    length    of  a    perpendicular,   from   the 
centre  of  the  vertical   shaft,  to  the  line  joining  the   points  of  the    brake    and   bell 
crank    to    which    the    link    was    attached,   was    9.743    feet  ;     the    effective    length 
of  the  vertical   arm   of   the    bell   crank,  was    4.5   feet,   and   of    the   horizontal   arm 
to   which   the   scale  was    attached,   5   feet;     consequently,   the    effective    length-   of 
the   brake   was 

9.743  X  5  ^  1Q826  feet 
4.5 

117.  COLUMN  15.     Quantity  of  water   passing    the   wheel,  in  cubic  feet   per  second. 
This   quantity  was   gauged   at  the  weir.     The   length  of  the  weir  was  13.998  feet; 
the  width   of    the   raceway  on   the   upstream   side   of    the   weir,  was   17  feet  ;    the 
crest  of  the  weir  was  11.14  feet  above  the  bottom  of  the  raceway.     The  quantity 
has  been   computed   by   the   formula 


determined  from   the   experiments   made,  in   1852,  at   the   Lower  Locks.      (See   art 
258.)     In  this  formula 

Q  =  ihe  quantity   in   cubic   feet   per  second. 

J  =  the  length   of  the   weir  =13.998  feet. 

n  =  the  number  of  end   contractions  =  2. 

h  =  the  depth   on   the   weir,  given  in   column   14. 


66 


EXPERIMENTS   ON  A   CENTRE-VENT  WATER-WHEEL, 


TABLE 

EXPERIMENTS  ON  THE  BOOTT 


1 

a 

3 

4 

5 

6 

7 

8 

9 

1O 

TIME. 

m-i-i 

ature  of 

J-Otal 
Dumber 

Useful  effect, 

the 

Height 

nf 

or  the 

No. 
of  the 

DATE, 
1849. 

water 
in  de- 
grees of 
Fahren- 
heit's 
ther- 

of the 
regu- 
lating 
gate, 
in 
nches. 

Beginning  of  the 
experiment. 

Ending  of  the 
experiment. 

LJUraiion 

of  the 
experi- 
ment, in 
seconds. 

OI 

revolu- 
tions 
of  the 
wheel 
during 
the 

iMimDcr  01 
revolutions 
of  the  wheel 
per  second. 

*v  eignc  1  11 
the  scale, 
hi  pounds 
avoirdupois. 

friction  of 
the  brake, 
in  pounds 
avoirdupois, 
raised  one 
foot  per 

experi- 

mome- 

experi- 

second. 

ment. 

ter. 

H. 

luin  . 

sec. 

H. 

lpin. 

sec. 

ment. 

1 

October  17.  A.M. 

54 

3 

10 

13 

19 

10 

17 

32 

253 

150 

0.59289 

575.56 

23211.8 

2 

u 

11 

30 

3.5 

11 

36 

42 

398.5 

350 

0.87829 

202.09 

12073.5 

3 

it                 it                 tt 

n 

It 

11 

46 

11 

11 

54 

15 

484 

350 

0.72314 

407.25 

20032.3 

4 

"       29,      " 

49.5 

tl 

11 

2 

17 

11 

19 

7 

1010 

550 

0.54455 

606.00 

22447.2 

5 

u           u           u 

U 

tl 

11 

33 

41 

11 

45 

22 

701 

350 

0.49929 

666.34 

22630.5 

6 

(i           a           (t 

u 

ft 

11 

59 

24 

0 

6 

49 

445 

200 

0.44944 

720.50 

22026.8 

7 

November  5,  p.  M. 

44 

tt 

4 

14 

47 

4 

21 

20 

393 

100 

0.25445 

931.87 

16129.1 

8 

"           7,  A.M. 

45 

tt 

9 

40 

19 

9 

48 

32 

493 

500 

1.01420 

0. 

0. 

9 

October  17,  P.M. 

53 

6 

2 

34 

15.5 

2 

44 

59 

643.5 

650 

1.01010 

334.06 

22952.9 

10 

u           u           u 

» 

tt 

2 

56 

6 

3 

5 

39 

573 

550 

0.95986 

441.22 

28807.9 

11 

ft               11               it 

u 

(t 

3 

17 

rr 

6 

3 

26 

1  f\ 

56 

OO  £\ 

590 

A  GO  K. 

550 

A  x.r\ 

0.93220 

A  OAAOA 

501.72 

Pi  co  J^n 

31814.1 

Q  A  A  *?  {*  A 

12 
13 

tt           it           it 

it 

u 

4 

50 

36 

5 

10 

4 

fi£i& 

13.5 

4JU.O 

817.5 

401) 

700 

U.UUUSJU 
0.85627 

Obi.oU 

656.59 

O44/D.U 
38243.0 

14 

"       29,      " 

'  50 

tt 

2 

4 

10 

2 

14 

58 

648 

450 

0.69444 

955.50 

45135.3 

15 

a        it        it 

U 

u 

2 

41 

41 

2 

52 

50 

669 

400 

0.59791 

1140.94 

46402.8 

16 

November  7,  A.M. 

45 

it 

9 

29 

27 

9 

37 

57 

510 

600 

1.17647 

0. 

0. 

17 

October  29,  P.M. 

50 

9 

3 

20 

41 

3 

27 

28 

407 

450 

1.10565 

263.00 

19779.8 

18 

tl               U               U 

it 

tt 

3 

33 

18 

3 

35 

49 

151 

150 

0.99338 

531.75 

35931.0 

19 

u          u          a 

it 

tt 

3 

36 

44 

3 

44 

8 

444 

400 

0.90090 

786.75 

48212.7 

20 

tt          tt          tt 

u 

tt 

3 

45 

5 

3 

54 

10.5 

545.5 

450 

0.82493 

1001.47 

56195.7 

21 

U               U               It 

u 

a 

3 

55 

14 

4 

6 

53.5 

699.5 

550 

0.78628 

1107.37 

59226.4 

22 

It               It               U 

ti 

a 

4 

21 

14 

4 

80 

10 

536 

400 

0.74627 

1205.00 

61168.8 

23 

tt               U               ft 

u 

(C 

4 

31 

A  1 

19 

4 

40 

C  -I 

31 

4   FT 

552 

C  /»i)    E 

400 

A  AA 

0.72464 

1259.16 

62065.4 

24 
25 

tt            ft            tt 

u 

a 

4 
4 

41 
54 

42 
35 

4 
5 

51 
4 

.5 
7.5 

502.5 
572.5 

400 
400 

0.71111 
0.69869 

1297.31 
1329.78 

62752.2 
63199.3 

26 

November  7,  A.M. 

45 

tt 

9 

19 

24 

9 

27 

22.5 

478.5 

600 

1.25392 

0. 

0. 

27 

November  5,  A.  M. 

44 

12 

9 

4 

34.5 

9 

13 

58.5 

564 

400 

0.70922 

1554.22 

74979.3 

28 

tt          tt       tt 

u 

ti 

9 

15 

10 

9 

21 

7.5 

357.5 

250 

0.69930 

1584.00 

75347.2 

29 

tt          tt      it 

tt 

ti 

9 

33 

15 

9 

39 

23 

368 

250 

0.67935 

1613.94 

74580.9 

30 

It                   tl           tt 

u 

ti 

9 

40 

37 

9 

48 

3.5 

446.5 

300 

0.67189 

1644.37 

75153.2 

31 

It                  It           It 

It 

tt 

10 

0 

3 

10 

7 

37.5 

454.5 

300 

0.66007 

1675.06 

75208.3 

32 

U                  It           It 

It 

tt 

10 

8 

54.5 

10 

16 

37 

462.5 

300 

0.64865 

1705.47 

75249.2 

33 

tt              tf         tt 

It 

ti 

10 

32 

31 

10 

41 

41 

550 

350 

0.63636 

1735.94 

75142.9 

34 

ft            It        tt 

tl 

u 

10 

43 

0 

10 

51 

0.5 

480.5 

300 

0.62435 

1768.41 

75103.3 

35 

It            It        tt 

u 

11 

11 

1 

53 

11 

10 

2 

489 

300 

0.61350 

1802.06 

75202.0 

36 

tt              tt         tt 

u 

11 

11 

11 

24 

11 

18 

26 

422 

250 

0.59242 

1836.19 

73993.4 

37 

"             6,    P.M. 

u 

11 

3 

31 

12 

3 

35 

20 

248 

0 

0. 

3155.34 

0. 

38 

a            tt        u 

u 

tt 

3 

40 

16 

3 

42 

22 

126 

0 

0. 

2797.27 

0. 

39 

"             7,   A.M. 

45 

tl 

9 

10 

57 

9 

18 

5 

428 

550 

1.28505 

0. 

0. 

1 

In  experiments  Nos.  8,  16,  26,  and  39,  the  brake  was  removed. 

In  experiment  No.  37,  the  weight  preponderated.    In  No.  38,  the  wheel  preponderated  (art.  77.) 


AT  THE  BOOTT  COTTON-MILLS. 


67 


VI. 

CENTRE- VENT  WATEIMVHEEL 


11 

13 

13 

14 

15 

16 

17 

18 

19 

20 

No. 

Height  of 

Ratio  of  the 

of 
the 
exper- 

Height of 
the  water 
above  the 

the  water 
below  the 
wheel, 
taken  in  the 

Total  fall 
acting  upon 
the  wheel. 

Depth  of 
water  on 
the  weir. 

Quantity  of 
water  passing 
the  wheel,  in 

Total  power  of 
the  water, 
in  pounds 
avoirdupois, 

Ratio  of  the 
useful  effect  to 
the  power 

Velocity  due 
to  the  fall 
acting  on 
the  wheel, 

Velocity  of 
the  exterior 
circumference 
of  the  wheel, 

velocity  of  the 
exterior 
circumference 
of  the  wheel, 
to  the  velocity 

iment. 

wheel. 

wheelpit. 

cubic  feet 
per  second. 

raised  one  foot 
per  second. 

expended. 

hi  feet  per 
second. 

in  feet  per 
second. 

due  to  the  fall 
acting  on  the 
wheel. 

Feet. 

Feet. 

Feet. 

Feet. 

1 

16.013 

1.410 

14.603 

1.2964 

67.532 

61493.4 

0.37747 

30.648 

17.393 

0.56750 

2 

16.036 

1.364 

14.672 

1.2619 

64.887 

59364.4 

0.20338 

30.721 

25.766 

0.83871 

3 

15.955 

1.387 

14.568 

1.2821 

66.432 

60347.0 

0.33195 

30.612 

21.214 

0.69301 

4 

15.558 

1.400 

14.158 

1.2845 

66.614 

58821.6 

0.38161 

30.178 

15.975 

0.52937 

5 

15.607 

1.410 

14.197 

1.2899 

67.029 

59351.7 

0.38129 

30.219 

14.647 

0.48470 

6 

15.563 

1.420 

14.143 

1.2881 

66.889 

59002.2 

0.37332 

30.162 

13.185 

0.43714 

7 

15.604 

1.360 

14.244 

1.2943 

67.368 

59858.1 

0.26946 

30.269 

7.465 

0.24661 

8 

15.573 

1.273 

14.300 

1.2115 

61.083 

54486.4 

0. 

30.329 

29.753 

0.98101 

9 

15.956 

1.668 

14.288 

1.5145 

84.998 

75732.8 

0.30308 

30.316 

29.633 

0.97746 

10 

15.930 

1.704 

14.226 

1.5308 

86.355 

76608.0 

0.37604 

30.250 

28.159 

0.93086 

11 

15.914 

1.717 

14.197 

1.5395 

87.080 

77093.2 

0.41267 

30.219 

27.347 

0.90496 

12 

15.923 

1.730 

14.193 

1.5467 

87.685 

77607.2 

0.44424 

30.215 

26.429 

0.87470 

13 

15.944 

1.750 

14.194 

1.5539 

88.285 

78143.8 

0.48939 

30.216 

25.120 

0.83134 

14 

15.581 

1.803 

13.778 

1.5762 

90.166 

77480.4 

0.58254 

29.770 

20.372 

0.68433 

15 

15.481 

1.875 

13.606 

1.5943 

91.697 

77812.1 

0.59634 

29.584 

17.540 

0.59291 

16 

15.451 

1.506 

13.945 

1.4180 

77.112 

67076.7 

0. 

29.950 

34.513 

1.15237 

17 

15.408 

1.890 

13.518 

1.6418 

95.762 

80736.3 

0.24499 

29.488 

32.436 

1.09997 

18 

15.323 

1.950 

13.373 

1.6734 

98.490 

82145.2 

0.43741 

29.329 

29.142 

0.99362 

19 

15.352 

1.983 

13.369 

1.6955 

100.418 

83728.2 

0.57582 

29.325 

26.429 

0.90125 

20 

15.413 

2.017 

13.396 

1.7184 

102.421 

85571.4 

0.65671 

29.354 

24.200 

0.82442 

21 

15.426 

2.047 

13.379 

1.7230 

102.825 

85800.0 

0.69029 

29.336 

23.066 

0.78629 

22 

15.418 

2.076 

13.342 

1.7308 

103.517 

86138.0 

0.71013 

29.295 

21.893 

0.74731 

23 

15.424 

2.102 

13.322 

1.7337 

103.769 

86218.8 

0.71986 

29.273 

21.258 

0.72620 

24 

15.465 

2.131 

13.334 

1.7328 

103.689 

86229.3 

0.72774 

29.286 

20.861 

0.71232 

25 

15.464 

2.160 

13.304 

1.7389 

104.229 

86483.7 

0.73077 

29.253 

20.497 

0.70067 

26 

15.417 

1.715 

13.702 

1.5981 

92.018 

78648.0 

0. 

29.688 

36.785 

1.23907 

27 

15.398 

1.998 

13.400 

1.8316 

112.525 

94057.5 

0.79716 

29.359 

20.806 

0.70868 

28 

15.434 

2.003 

13.431 

1.8367 

112.987 

94662.2 

0.79596 

29.393 

20.515 

0.69796 

29 

15.321 

1.990 

13.331 

1.8320 

112.562 

93603.9 

0.79677 

29.283 

19.929 

0.68058 

30 

15.369 

1.991 

13.378 

1.8368 

112.996 

94296.4 

0.79699 

29.335 

19.711 

0.67193 

31 

15.367 

1.981 

13.386 

1.8377 

113.071 

94415.2 

0.79657 

29.343 

19.364 

0.65990 

32 

15.369 

1.986 

13.383 

1.8387 

113.164 

94471.1 

0.79653 

29.340 

19.029 

0.64856 

33 

15.336 

1.980 

13.356 

1.8379 

113.090 

94219.0 

0.79753 

29.311 

18.668 

0.63692 

34 

15.362 

1.981 

13.381 

1.8443 

113.673 

94881.9 

0.79154 

29.338 

18.316 

0.62431 

35 

15.385 

1.980 

13.405 

1.8511 

114.293 

95571.2 

0.78687 

29.364 

17.998 

0.61291 

36 

15.292 

1.971 

13.321       1.8476 

113.969 

94703.1 

0.78132 

29.272 

17.379 

0.59371 

37 

15.442 

1.905 

13.537  •     1.8087 

110.454 

93270.1 

0. 

29.508 

0. 

0. 

38 

15.477 

1.902 

13.575       1.8072 

110.325 

93422.4 

0. 

29.550         0. 

0. 

39 

15.415 

1.819 

13.596 

1.6884 

99.795 

84635.0 

0. 

29.573  i     37.698 

1.27477 

68  EXPERIMENTS   ON   A   CENTRE-VENT  WATER-WHEEL, 

118.  The   results   of  the   experiments  in   table   VI.,  are   represented  by  a  sys- 
tem of  coordinates  at  figure  1,  plate  IX. ;  —  the  relative  velocities,  given  in  column 
20,  are   taken   for  the   abscissas,  and   the   corresponding   ratios   of  the   useful   effects 
to   the   powers  expended,  given  in   column   17,  are   taken   for  the   ordinates.      The 
numbers  on  the   figure   refer    to    the    experiments   in   table   VI.,  which   the   several 
points    represent ;  —  the    points    not    numbered    represent    some    experiments     not 
reported,   in   consequence   of    an    imperfection    in    the    gauge    of    the   quantity   of 
water   discharged,  owing  to   a   defective  arrangement  of  the  grating.     These  experi- 
ments  have   been   corrected    by   a    comparison  with   those   that  are   reported ;    not- 
withstanding   this    correction,    however,   they   ought    not    to    be    considered    as    of 
equal   value    with    those    reported    in    table    VI.      In    the   figure,   the   points   repre- 
senting the  latter  experiments,  are  connected  by  full  lines;    the   points  representing 
the    experiments    considered    imperfect,   are   connected    by   broken   lines.      The    line 
AB  represents   the    experiments   reported,  that  were   made  with  the  regulating  gate 
fully   raised ;    the   line   CD,  the   experiments   with    the    gate    raised    three    quarters 
of    its   full    height ;     EF,   the    experiments   with    the    gate   raised   a   half,   and    GIf, 
the   experiments   with   the   gate    raised   one    quarter   of  its   full   height.      It   will   be 
seen    that   the   maximum   coefficient   of  effect,  with    the   gate   fully    raised,   is   given, 
when   the  outside    of   the   wheel   is   moving   with   a   velocity   equal   to    about   sixty- 
seven   per   cent,  of  that   due   to   the  fall   acting  upon  the  wheel,  at  which  velocity, 
the   useful    effect   is  very  nearly  eighty  per   cent,  of  the    total   power   of  the  water. 
The   coefficient   of  effect   diminishes   rapidly   as  the   regulating   gate   is   lowered,   ;md 
the    maximum    is   also   found   at   a   slower   speed ;     thus,    when   the    gate    is    raised 
three   inches,   or  one    quarter   of  its   full   height,   the   maximum   coefficient   of  effect 
is   thirty-eight  per  cent,  of  the  power  expended ;    which  is  given  when  the  outside 
of  the   wheel   is   moving  with   a   velocity   about   one    half  of    that   due    to   the   fall 
acting  upon  the  wheel. 

119.  AB  CD,   figure    2,   plate    IX.,   represents    the   path   of    the   water   as    it 
passed   through   one   of  the    apertures   of  the   wheel,  in  experiment  30,  according  to 
the   hypothesis   in   art.    83 ;     the   steps   in   the    calculation    for   which,    are   given   in 
table  VII.     In  the  formula 


we  have  for  this  case, 

0  =  the  ordinate   measured   on  the   arc   of  a  circle  the  radius  of  which   is  R ; 

its  several  values  are  given  in  column  10. 
R  =  the    distance    from    the   centre    of   the   wheel   for  which    the    ordinate    is 


AT  THE,  BOOTT   COTTON-MILLS.  69 

computed  ;  —  its  several  values  are   given   in  inches,  in  column  1  ;  —  to 
compute   the   value   of  0  in   feet,  R  must  be  taken  in  feet. 
w  =  the   angular  velocity.      In   experiment   30,  the  velocity   of  the   outside   of 
the   wheel   was   19.711    feet   per    second,   and   the   radius   of   the   out- 
side  of  the   wheel   is   4.669   feet,   consequently, 

_  19.711  _    49917 
-T669  -    42217- 

AH=  the  volume  of  that  part  of  the  space  between  two  adjacent  buckets, 
included  between  the  outside  of  the  wheel  and  the  radius  R'  ;  —  its 
several  values  are  given  in  column  9. 

(>"  =  the  quantity  of  water  discharged,  per  second,  by  each  orifice  in  the  wheel. 
In  experiment  30,  we  have,  by  table  VI.,  the  total  quantity  dis- 
charged =  112.996  cubic  feet  per  second,  and  as  there  are  forty 
orifices,  we  have 

112.996 


=  2.8249. 


In  figure  2,  plate  IX.,  the  buckets  and  guides  are  drawn  to  a  scale  one 
fourth  the  full  size;  —  the  radius  of  the  arc  AB  =  R  =  5Q.Q28  inches.  To  find 
the  limit  of  the  stream  on  the  side  B  C,  the  arcs  IF,  KH,  etc.,  N  C,  are  drawn 
with  the  radii  55  inches,  54  inches,  etc.,  47.922  inches;  —  the  arcs  E  F,  G  H,  etc., 
0  C,  being  taken  from  column  10,  equal  to  0.415  feet,  0.796  feet,  etc.,  2.748  feet  ; 
the  points  B,  F,  If,  etc.,  C,  being  connected  by  suitable  lines,  determine  the  limit  of 
the  stream  on  that  side.  The  limit  of  the  stream  on  the  other  side  is  found  by 
making  the  arcs  FL  =  UI,  HM=  G  K,  etc.,  CD=  ON;  —  the  points  A,  L,  M,  etc., 
D,  being  connected  by  suitable  lines,  determine  the  limit  of  the  stream  on  that  side. 

By  an  examination  of  figure  2,  it  will  be  seen,  that  the  section  of  the  stream 
just  after  it  has  entered  the  wheel,  is  sensibly  greater  than  the  section  of  the 
stream  as  it  leaves  the  guides,  and  that,  consequently,  if  the  stream  flowed 
according  to  the  hypothesis,  there  must  have  been  a  sudden  change  in  the 
velocity  of  the  water,  causing  a  shock,  which,  according  to  the  common  theory, 
implies  a  loss  of  power.  This  indicates  a  defect  in  the  design;  nevertheless,  the 
success  attending  this  first  essay,  on  a  large  scale,  of  a  centre-vent  water-wheel,  in 
which  due  regard  has  been  paid  to  accuracy  of  construction  and  perfection  of 
workmanship,  guided  by  such  light  as  the  present  imperfect  theories  can  afford, 
ought  to  encourage  us  to  hope,  that,  when  it  has  received  the  same  degree  of 
attention  as  the  turbine,  it  will  not  be  much  behind  that  celebrated  motor,  in 
its  economical  use  of  water. 


70 


EXPERIMENTS   ON  A   CENTRE-VENT   WATER-WHEEL. 


TABLE    VII. 


1 

a 

3 

4 

5 

6 

7 

8 

9 

1O 

1 

Ordinate* 

^JT 

in  feet,  to 

Value  of 

Areas  in 

Areas  in 

of  the  areas  of 

Correction 

Corrected 

Mean 

Volumes 

be  meas- 

A and 

square  inches, 

square 

the  complete 

for  the 

areas  of  the 

height 

Volumes  of 

between  R 

ured  on 

successive 

of  circles  of 

inches,  of  the 

rings  in  the 

thickness  of 

partial  rings, 

of  the 

the  partial 

and  the 

arcs  of  the 

values  of 

the  radii  in  the 

complete 

preceding 

the  bucket,  in 

in  square 

partial 

rings,  In 

successive 

corre- 

.ft', in 
inches. 

preceding 

column. 

rings. 

column,  in 
square  feet. 

square  feet. 

feet. 

rings, 
in  feet. 

cubic  feet. 

values  of  R^ 
in  cubic  feet. 

sponding 
radii  in 

column  1. 

56.028 

9861.890 

55.000 

9503.318 

358.572 

0.06225 

0.00168 

0.06057 

1.001 

0.06063 

0.06063 

0.415 

54.000 

9160.884 

342.434 

0.05945 

0.00210 

0.05735 

1.008 

0.05781 

0.11844 

0.796 

53.000 

8824.734 

336.150 

0.05836 

0.00227 

0.05609 

1.021 

0.05727 

0.17571 

1.160 

52.000 

8494.866 

329.868 

0.05727 

0.00262 

0.05465 

1.042 

0.05695 

0.23266 

1.507 

51.000 

8171.282 

323.584 

0.05618 

0.00304 

0.05314 

1.070 

0.05686 

0.28952 

1.839 

50.000 

7853.982 

317.300 

0.05509 

0.00386 

0.05123 

1.105 

0.05661 

0.34613 

2.155 

49.000 

7542.964 

311.018 

0.05400 

0.00561 

0.04839 

1.147 

0.05550 

0.40163 

2.451 

48.750 

7466.191 

76.773 

0.01333 

0.00168 

0.01165 

1.177 

0.01371 

0.41534 

2.522 

48.500 

7389.811 

76.380 

0.01326 

0.00181 

0.01145 

1.190 

0.01363 

0.42897 

2.591 

48.250 

7313.824 

75.987 

0.01319 

0.00202 

0.01117 

1.204 

,  0.01345 

0.44242 

2.659 

47.922 

7214.723 

99.101 

0.01721 

0.00252 

0.01469 

1.221 

0.01794 

0.46036 

2.748 

PART    II. 

EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS,  AND  IN 
SHORT  RECTANGULAR  CANALS. 


EXPERIMENTS   ON  THE   FLOW  OF   WATER  OVER  WEIRS. 

120.  THE  laws  governing  the  flow  of  water  over  weirs,  have  received  the 
attention  of  several  distinguished  engineers  and  men  of  science,  among  whom  may 
be  named  Smeaton  and  Brindley  in  England;  Du  Buat,  Navier,  D'Aubuisson, 
Castel,  Poncelet,  Lesbros,  and  Boileau,  in  France ;  and  Eytelwein  and  Weisbach  in 
Germany.  A  great  number  of  experiments  have  been  made  and  recorded ;  the 
earlier  ones  rude  and  imperfect ;  the  later  ones,  particularly  those  by  Poncelet, 
Lesbros,  and  Boileau,  with  a  perfection  of  apparatus  previously  unknown. 

There  has  been  in  this  branch  of  hydraulics,  as  well  as  in  others,  a  steady 
advance  with  the  accumulation  of  experiments  and  the  improvement  of  the  means 
of  observation ;  the  result,  however,  of  these  numerous  labors,  is  far  from  satisfac- 
tory to  the  practical  engineer.  On  a  careful  review  of  all  that  has  been  done, 
he  finds  that  the  rules  given  for  his  use,  are  founded  on  the  single  natural  law 
governing  the  velocity  of  fluids,  known  as .  the  theorem  of  Torricelli ;  omit- 
ting, in  consequence  of  the  extreme  complexity  of  the  subject,  all  consideration 
of  many  other  circumstances,  which,  it  is  well  known,  materially  affect  the  flow  of 
water  through  orifices.  He  finds  also  that  it  has  been  attempted  to  correct  the 
theoretical  expression  thus  found,  by  coefficients  obtained  by  comparing  the  results 
derived  from  it,  with  those  furnished  by  experiment ;  but  when  he  comes  to 
investigate  these  experiments,  even  after  rejecting  all  excepting  those  made  with 
the  greatest  care,  and  with  apparatus  capable  of  insuring  the  greatest  precision, 
he  finds  such  discordances  in  the  resulting  coefficients,  that  he  loses  all  hope  of 
arriving  at  correct  results  when  he  applies  them  on  the  great  scale.  They  will 
undoubtedly  furnish  sufficiently  accurate  results,  if  the  apparatus  used  is  a  repro- 


72  EXPERIMENTS   ON   THE   FLOW  OF   WATER  OVER  WEIRS. 

duction,  both  in  form  and  dimension,  of  that  used  in  the  experiments ;  but  this 
is  seldom  attainable,  the  experiments  having  been  made  on  such  a  minute  scale. 
Boileau,*  in  discussing  the  various  formulas  that  have  been  proposed,  points  out 
many  of  their  defects,  and  has  himself  proposed  a  new  one,  coupled,  however, 
with  some  special  conditions  in  the  form  of  the  weir,  and  the  mode  of  taking 
the  depth  upon  the  sill. 

No  correct  formula  for  the  discharge  of  water  over  weirs,  founded  upon  natu- 
ral laws,  and  including  the  secondary  effects  of  these  laws,  being  known,  we  must 
rely  entirely  upon  experiments,  taking  due  care  in  the  application  of  any  formula 
deduced  from  them,  not  to  depart  too  far  from  the  limits  of  the  experiments  on 
which  it  is  founded. 

Engineers  have  generally  agreed  that  the  most  convenient  form  of  weir  for 
gauging  streams  of  water,  is  one  which  is  cut  in  a  vertical  plane  side  of  a  reser- 
voir, the  sill  being  horizontal,  the  sides  vertical,  and  the  contraction  complete.  In 
order  that  the  contraction  may  be  complete,  the  sill  and  sides  of  the  weir  must 
be  so  far  removed  from  the  bottom  and  lateral  sides  of  the  reservoir,  that  they 
may  produce  no  more  effect  upon  the  discharge,  than  if  they  were  removed  a 
distance  indefinitely  great ;  also,  the  aperture  must  be  effectively  the  same,  as  if 
cut  in  a  plate  having  no  sensible  thickness.  The  condition  relating  to  the  dis- 
tance of  the  bottom  and  sides  of  the  reservoir,  can  seldom  be  strictly  complied 
with,  when  gauging  large  streams  of  water ;  it  is  found,  however,  that,  when  the 
sill  is  at  a  height  above  the  bottom  of  the  reservoir  not  less  than  twice  the 
height  of  the  water  above  the  sill,  and  the  sides  are  removed  a  distance  at  least 
equal  to  the  height  above  the  sill,  a  correction  free  from  serious  error  can  usually 
be  made  for  the  effect  of  the  velocity  of  the  .water  approaching  the  weir.  The 
condition  that  the  aperture  shall  be  effectively  the  same  as  if  cut  in  a  plate 
having  no  sensible  thickness,  is  usually  more  easily  complied  with.  The  effect  of 
the  contraction  is  such,  that  the  water  has  a  strong  tendency  to  leave  the  bottom 
and  sides  of  the  aperture  for  a  certain  distance,  and  to  touch  the  aperture  only 
at  the  upstream  edge ;  if,  however,  the  thickness  of  the  plank  or  other  material, 
exceeds  a  certain  amount,  (depending  upon  the  depth  flowing  over,)  the  water 
will  follow  the  top  of  the  plank ;  in  this  case,  all  that  is  requisite  is,  to  cut 
away  the  downstream  side  of  the  weir  at  an  angle  of,  say,  forty-five  or  sixty 
degrees  with  the  horizontal ;  leaving  horizontal,  only  a  small  part  of  the  thick- 


*  Jaugeage  des  cours  d"eau  a  faille  ou  a  moyenne  section  by  M.  P.  Boileau  (Paris :   1850)  ;   or  Journal 
de  I'Ecok  Polytechnique,  No.  xxxiii. 


EXPERIMENTS   ON   THE   FLOW  OF   WATER   OVER  WEIRS.  73 

ness  of  the  sill.  It  is  essential,  however,  that  the  corners  of  the  sill  and  sides 
of  the  weir  presented  to  the  stream,  should  be  full  and  sharp,  and  not  rounded 
or  bevelled  in  any  degree. 

121.  Two  modes  present  themselves  for  studying,  experimentally,  the  laws 
governing  the  discharge  of  water  over  weirs.  First,  that  which  has  been  uni- 
formly adopted  heretofore,  namely,  to  obtain  by  direct  measurement  the  quantity 
of  water  discharged  in  a  given  time,  through  an  aperture  of  known  dimensions  ; 
this  is  evidently  the  only  mode  of  resolving  the  question  completely.  To  per- 
form the  experiments,  however,  upon  a  scale  of  magnitude  corresponding  to  the 
ordinary  practical  applications,  usually  requires  an  apparatus  of  great  cost,  and  such 
as  is  beyond  the  reach  of  most  experimenters.  The  great  difficulty  is,  to  obtain 
a  suitable  basin,  in  which  to  make  the  direct  measurement  of  the  quantity  dis- 
charged by  the  weir. 

The  second  mode  dispenses  with  a  direct  measurement  of  the  quantity.  If 
we  have  two  weirs  of  the  same  form,  but  of  different  lengths,  and  we  know  that 
the  quantities  of  water  discharged  by  them,  in  certain  circumstances,  are  equal  ; 
knowing  also  the  depth  upon  the  sill  of  each  weir,  we  have  the  data  for  an 
equation  by  which  one  unknown  quantity  may  be  determined.  Neither  the 
coefficient  of  contraction,  nor  the  absolute  discharge  can,  however,  be  obtained 
by  such  an  equation. 

'  122.  The  discharge  over  weirs  is  commonly  assumed  to  vary  as  the  square 
root  of  the  third  power  of  the  depth  ;  let  us  suppose  it  to  be  unknown,  and 
equal  to  a. 

Suppose  also  I  the  length,  and  h  the  depth,  on  one  of  the  weirs  ;  and  I'  and 
//  the  corresponding  dimensions  for  the  other  weir  ;  O,  a  constant  coefficient  ;  Q, 
the  quantity  which,  by  hypothesis,  is  the  same  for  both  weirs.  Assuming,  accord- 
ing to  the  common  formula,  that  the  quantity  is  proportional  to  the  length  of 
the  weir,  we  have 


Q=  Cl'h'a; 
consequently, 

Clh°=  Ol'K'i 


taking  the  logarithms,  we  have 

a  (Log.  h  —  Log.  A')  =  Log.  I'  —  Log.  I ; 
10 


74  EXPERIMENTS  ON   THE   FLOW   OF   WATER  OVER   WEIRS. 

therefore, 

_  Log.  /' —  Log.  I 
"Log.  h—  Log.  h" 

We  can  thus,  by  means  of  two  experiments,  determine  the  power  of  the 
depth  which  will  lead  to  identical  quantities  in  the  computed  discharge  of  the 
two  weirs. 

123.  It   is   assumed   in   the   above   equations,   that   the   quantity   discharged   by 
a  weir   is   directly  proportioned   to   its   length;    this,  in  weirs   having  complete  con- 
traction,   is,   however,   known  not  to   be   true,   in    consequence    of    the    contraction 
which   takes   place   at   the   ends   of  the   weir.      This   contraction    diminishes   the   dis- 
charge.     When   the   weir  is  of  considerable   length  in  proportion   to   the   depth   of 
the  water   flowing   over,  this   diminution   is   evidently  a  constant  quantity,  whatever 
may  be   the   length,   provided   the    depth   is   the  same ;    we   may,   therefore,   assume 
that   the    end    contraction    effectively    diminishes    the    length    of    such   weirs,   by   a 
quantity   depending  only  upon   the   depth   upon   the   weir.     It  is  evident  that   the 
amount   of    this    diminution    must  increase   with  the   depth ;    we   are   unable,   how- 
ever,  in    the    present  state   of    the   science,   to   discover  the   law   of   its   variation ; 
but   experiment    has    proved    that    it    is  very   nearly   in    direct   proportion   to    the 
depth.      As  it   is   of    great   importance,   in   practical    applications,   to   have   the   for- 
mula as  simple  as  possible,   it  is  assumed    in    this   work   that   the   quantity  to  be 
subtracted   from    the    absolute   length    of   a   weir    having    complete   contraction,  to 
give   its   effective   length,  is  directly  proportional  to  the   depth.     It  is   also  assumed 
that  the   quantity   discharged  by  weirs  of  equal  effective   lengths,   varies  according 
to   a   constant  power   of  the   depth.      There   is   no   reason   to   think   that   either   of 
these   assumptions  is  perfectly  correct ;    it  will  be   seen,  however,  that  they  lead  to 
results  agreeing  very  closely  with  experiment. 

124.  The   formula   proposed  for  weirs   of  considerable   length   in   proportion  to 
the   depth  upon  them,  and  having  complete  contraction,  is 

Q=C(l— bnh)h°;* 
in  which 

Q  =  the  quantity  discharged  in  cubic  feet  per  second. 
C=  a  constant  coefficient. 
l=ihe  total  length  of  the  weir  in  feet. 
b  =  a  constant  coefficient. 


*  This  formula  was  first  suggested  to  the  author  by  Mr.  Boyden,  in  1846. 


EXPERIMENTS   ON   THE   FLOW  OF   WATER  OVER  WEIRS.  75 

H  =  the  number  of  end  contractions.  In  a  single  weir  having  complete  con- 
traction, n  always  equals  2,  and  when  the  length  of  the  weir  is  equal  to 
the  width  of  the  canal  leading  to  it,  n  =  0. 

h  =  the  depth  of  water  flowing  over  the  weir,  taken  far  enough  upstream  from 
the  weir,  to  be  unaffected  by  the  curvature  in  the  surface  caused  by  the 
discharge. 

a  =  a  constant  power. 

The  coefficient  C  can  be  determined  only  from  experiments  in  which  the 
actual  discharge  is  known ;  the  constants,  a  and  b  can,  however,  be  determined 
without  knowing  the  actual  discharge  in  any  particular  case. 

It  has  been  stated  that  the  proposed  formula  is  applicable  only  to  weirs 
having  a  considerable  length  in  proportion  to  the  depth  of  water  running  over 
them.  It  is  found  by  experiment  that,  when  the  length  equals  or  exceeds  three 
times  the  depth,  the  formula  applies ;  but  in  lengths  less  than  this  in  proportion 
to  the  depth,  the  formula  cannot  be  used  with  safety;  the  error  increasing  as  the 
relative  length  of  the  weir  diminishes. 

It  is  evident,  from  the  construction  of  the  formula,  that  it  cannot  be  of  gen- 
eral application.  The  factor  I — bnh  represents  the  effective  length  of  the  weir; 
if  l=bnh  this  effective  length  becomes  0,  and  the  formula  would  give  0  for 
the  discharge,  which  is  absurd;  similarly,  if  Inky  I,  the  discharge  given  by  the 
formula  would  be  negative.  In  weirs  of  very  short  length  in  proportion  to  the 
depth,  the  effect  of  the  end  contraction  cannot  be  considered  as  independent  of 
the  length.  The  end  contraction  influences  the  discharge  to  a  certain  distance,  A, 
from  the  end  of  a  weir ;  if  the  whole  length  of  the  weir  is  greater  than  2  A, 
the  effect  of  the  end  contraction  is  independent  of  the  length ;  but  if  the  length 
is  less  than  2  A,  the  whole  breadth  of  the  stream  is  affected  in  its  flow  by  the 
end  contractions,  and,  consequently",  the  proposed  formula  would  not  apply. 

In  practical  applications,  this  will  seldom  be  an  inconvenience,  as  it  is  nearly 
always  practicable  so  to  proportion  the  weir,  that  the  length  may  not  be  less 
than  three  times  the  depth  upon  it;  if,  however,  there  is  no  end  contraction,  the 
proportion  of  the  length  to  the  depth  is  not  material. 

125*  The  author  has  made  numerous  experiments  on  the  discharge  of  water 
over  weirs,  according  to  each  of  the  methods  described  above. 

First,  those  at  the  Tremont  Turbine,  and  at  the  centre-vent  water-wheel  for 
moving  the  guard  gates  of  the  Northern  Canal.  In  none  of  these  experiments 
has  any  attempt  been  made  to  measure  the  absolute  quantities  flowing  over  the 
weirs ;  but  simply  to  cause  quantities  of  water  known  to  be  equal,  to  pass  over 


76  EXPERIMENTS   ON  THE   FLOW  OF   WATER  OVER   WEIRS. 

weirs  of  different  dimensions,  noting  the  depth  of  water  and  length  of  weir  in 
each  case.  From  these  data,  as  is  explained  above,  certain  factors  in  the  formula 
can  be  determined. 

Second,  those  at  the  Lower  Locks,  in  which  the  absolute  quantities  passing 
over  weirs  of  known  dimensions,  were  measured  directly. 

As  each  of  these  three  sets  of  experiments  were  made  with  different  appara- 
tus, they  will  be  described  separately. 


EXPERIMENTS  MADE  AT  THE  TREMONT  TURBINE,  ON  THE  FLOW  OF  WATER  OVER  WEIRS. 

126.  The  apparatus  constructed  to  gauge  the  water  discharged  by  the  Tremont 
Turbine,  with   some    modifications,   was   used   for   the    experiments   on   the   discharge 
over  the  weir;    for  a  general  description  of  this  apparatus,  see  arts.  44,  45,  and  46. 

The  experiments  consisted  in  allowing  a  quantity  of  water,  of  unknown  vol- 
ume, to  enter  the  wheelpit,  through  the  turbine,  the  regulating  gate  of  which 
was  sufficiently  opened  for  the  purpose.  This  volume  of  water  was  then  caused 
to  flow  over  weirs  of  different  dimensions,  and  the  corresponding  depth  on  the 
weir,  assumed  by  the  water  in  each  experiment,  was  noted  after  the  water  had 
arrived  at  a  uniform  state. 

The  experiments  are  divided  into  series,  in  each  of  which  the  regulating  gate 
was  unchanged  throughout,  so  that  the  apertures  through  which  the  water  entered 
the  wheelpit  remained  constant  during  each  series. 

Some  variations  necessarily  occurred  in  the  head  acting  upon  these  orifices ; 
they  were  small,  however,  when  compared  to  the  whole  head.  The  depths  on 
the  weir  have  been  reduced,  according  to  well-known  principles,  to  what  they 
would  have  been  if  the  head  had  been  constant.  The  leakage  of  the  wheelpit 
also  rendered  another  small  correction  necessary.  After  the  corrections  are  made, 
we  have  in  each  series  a  collection  of  experiments  in  which  the  quantity  dis- 
charged is  the  same,  and  we  have  also  the  requisite  dimensions  of  the  different 
weirs.  These  data,  if  perfectly  accurate,  are  sufficient  to  enable  us  to  determine, 
in  the  proposed  formula  for  the  discharge,  the  values  of  the  constants  a  and  b. 
It  is  not  to  be  presumed,  however,  that  the  data  are  perfectly  correct,  but  we 
can,  at  any  rate,  find  the  values  of  a  and  b  that  will  give  the  most  uniform 
results  to  the  computed  discharges  in  all  the  experiments  in  a  series ;  the  actual 
discharge  being,  by  hypothesis,  a  constant  quantity. 

127.  Some    additions    to   the   apparatus   used   in  the   experiments  on   the   tur- 
bine  were   made   for   the  weir   experiments.      The  partitions,  represented   by  figures 


EXPERIMENTS   ON  THE   FLOW  OF   WATER  OVER    WEIRS.  77 

5,  6,  and  7,  plate  V.,  were  provided  for  the  purpose  of  shortening  or  subdividing 
the  weir.  They  were  made  of  wood,  faced  on  part  of  one  side  with  plates  of 
sheet-iron  a,  T3g  of  an  inch  in  thickness ;  the  width  b  c  was  about  1.5  feet ;  the 
iron  plate  was  two  inches  less.  One  side  of  the  timber  P,  figure  2,  was  in  the 
same  vertical  plane  as  the  upstream  edge  of  the  weir  H.  When  the  partitions 
were  placed  upon  the  weir,  the  top  of  them  was  supported  by  the  timber  P, 
and  the  bottom  by  the  plate  of  iron  a,  which  rested  against  the  weir..  Flash- 
boards,  represented  by  figures  8,  9,  and  10,  plate  V.,  were  also  provided  to  close 
up  portions  of  the  weir;  these,  together  with  the  partitions,  were  maintained  in 
their  respective  positions,  simply  by  the  pressure  of  the  water  against  them. 
Wherever  leaks  appeared  at  the  joints  of  the  partitions  or  flashboards,  they  were 
stopped  with  great  ease  and  effect,  by  a  little  dough  made  of  unbolted  Indian 
meal,  a  handful  of  which  was  drawn  over  the  upstream  side  of  the  joints ;  of 
course  the  orifices  closed  in  this  manner  were  very  minute.  In  plate  X.,  all  the 
modifications  of  the  weir  produced  by  changing  the  partitions  and  flashboards,  are 
represented ;  the  several  figures  are  referred  to  in  column  8,  table  X.  In  the 
greater  number  of  the  experiments,  two  or  more  spaces  were  used  at  the  same 
time ;  they  were  always  of  very  nearly  equal  length,  so  that  the  length  of  each 
may  be  obtained  by  dividing  the  whole  length  of  the  weir  given  in  column  6 
by  half  the  number  of  end  contractions  given  in  column  7. 

,The  brackets  N,  figures  1  and  2,  plate  V.,  were  placed  on  the  downstream 
side  of  the  weir,  to  support  a  board  on  which  to  stand  for  the  purpose  of 
adjusting  the  partitions  and  flashboards.  The  top  of  the  board  was  about  9.5 
inches  below  the  top  of  the  weir.  In  some  of  the  experiments,  a  part  of  the 
sheet  of  water  fell  upon  this  board ;  in  experiment  50  it  was  moved  nearer  to 
the  weir,  so  that  the  entire  sheet  of  water  fell  upon  it,  but  without  producing 
any  sensible  effect  upon  the  discharge.  In  experiment  51,  a  three  inch  plank 
was  placed  on  the  top  of  the  board,  as  is  represented  by  the  dotted  lines  at 
0,  figure  2,  plate  V. ;  the  effect  of  this  obstruction,  as  indicated  by  the  increased 
depth  on  the  weir  as  measured  by  the  hook  gauge,  was,  to  diminish  the  discharge, 
with  the  same  depth  on  the  weir,  about  foVo^ 

It  is  to  be  regretted  that  the  casting  forming  the  sill  of  the  weir,  was  not 
planed  on  its  whole  height  on  the  side  II Q,  figure  4,  plate  V.  When  the  weir 
was  erected  no,  thought  was  entertained  of  using  it  for  these  experiments, 
requiring,  as  they  do,  to  be  of  value,  to  be  free  from  all  disturbing  causes.  The 
disturbance  caused  by  the  projection  at  I,  can,  however,  have  been  scarcely  sensible. 

128.  The  data  furnished  by  observation,  together  with  the  necessary  reduc- 
tions, and  the  results  deduced  from  them,  are  contained  in  table  X.  Most  of 


78  EXPERIMENTS   ON  THE   FLOW   OF   WATER  OVER   WEIRS. 

the  columns  are  sufficiently  explained  by  the  respective  headings  ;    several  of  them, 
however,  require  further  explanation. 

129.  COLUMN  11.     Fall  affecting  the   leakage   of  the  wheelpit.     This  is  obtained   by 
adding  together  the  corresponding  numbers  in  columns  9  and  10. 

130.  COLUMN  12.     Depth    of  water   on  the    weir    corrected  for    the    leakage   of    the 
wheelpit.     This  is  obtained  in  the  following  manner. 

It  was  clear,  from  the  construction  of  the  wheelpit,  (art.  23,)  that  nearly  the 
whole  of  the  leakage  passed  through  the  wooden  flooring,  and  that  all  the  orifices 
through  which  it  passed  were  constantly  below  the  surface  of  the  lower  canal. 
In  the  construction  of  the  wheelpit,  no  particular  precautions  were  taken  to  pre- 
vent a  free  communication  from  the  bottom  of  the  wooden  flooring  to  the  lower 
canal  ;  and  as  the  amount  of  the  leakage  was  very  small,  and  the  material,  fine 
sand  free  from  large  springs,  it  is  clear  that  the  water  could  have  had  no 
appreciable  obstruction  after  passing  through  the  flooring,  except  from  the  pressure 
of  the  water  in  the  lower  canal.  This  being  the  case,  the  amount  of  the  leak- 
age would  depend  upon  the  head;  or,  in  other  words,  upon  the  height  from  the 
surface  of  the  water  in  the  wheelpit,  to  the  surface  of  the  water  in  the  lower 
canal.  Let 


j£  =  the   quantity  of  water  leaking  out  of  the  wheelpit,  in  cubic  feet   per 

second. 

A,  A',  A',  etc.  =  the   areas   of  the   several   orifices  through   which   the   water  passed. 
C,C',C",  etc.  :=  the   corresponding   coefficients  of  contraction. 

A:=the  head,  or  the  height  from  the  surface  of  the  water  in  the  wheel- 
pit,  to  the  surface  of  the  water  in  the  lower  canal.  This  head 
applies  to  all  the  orifices,  as  they  are  all  below  the  surface  of 
the  water  in  the  lower  canal. 


L  =  CA  \l  2ff h  +  C'A  \l  2 g h  -f  C"A'\f2gh  +  etc. ; 

L  =  ( CA  -f-  C'A'  +  C"A"+  etc.,)  V2gJ. 

The   areas  A,  A,  A",  etc.,  are   constant,   as   are   also   the   coefficients  C,  C',  C",  etc.,  the 
variations  in  the  head  not  being  very   great.      Let 

c  =  CA  +  C'A'  +  C"A"  +  etc. : 
then 


EXPERIMENTS   ON   THE   FLOW   OF   WATER  OVER   WEIRS.  79 


The  factor  c  \Zg,  being  constant,  can  be  determined  by  an  experiment  in  which 
L  and  h  are  known.  To  determine  this  constant,  the  following  experiment  was 
made. 

The  weir  was  closed  up  by  the  flashboards,  and  made  tight  in  the  usual 
manner,  so  that  no  appreciable  quantity  passed  over  the  weir  ;  the  head  gate 
was  closed,  and  the  small  quantity  leaking  through  it  was  caught  in  the  leak 
box  and  carried  over  the  weir  in  the  leak  pipe  (art.  24).  The  water  in  the 
wheelpit  having  then  no  supply,  its  surface  began  to  lower,  in  consequence  of 
the  leakage  through  the  floor  ;  while  thus  falling,  the  following  observations  were 
made. 

February  5,  1851,  at   10",  20',  30",  A.M.,  the   height  of  the  water 

in  the  wheelpit  above  the  top  of  the  weir,  was     .....     0.596  feet. 
And  at   llh,  1',  46",  A.M.,  the  height  was     .........     0.396     « 

Consequently  the   surface   of  the   water   in   the   wheelpit   lowered 

in   2476"      ...................     0.200  feet. 

The  area  of  the  surface  of  the  water  in  the  wheelpit,  after  making  the 
proper  deductions,  was  about  506  square  feet  ;  consequently, 

T        506X0.2  ,.      „  , 

=  —  2476     ==  O-O^O^  cubic  feet  per  second. 

During  the  interval  of  2476  seconds,  the  mean  height  of  the  water  in  the 
lower  canal  was  1.2316  feet  below  the  top  of  the  weir,  and  the  mean  height  in 
the  wheelpit,  during  the  same  period,  was  0.496  feet  above  the  top  of  the  weir, 
then 

h  =  1.2316  +  0.4960  =  1.7276  feet. 

^ 

Substituting  these  values  of  L  and  h  in  the  equation 


we  have 

=  0.03112: 


consequently, 

Z  =  0.03112  \fh. 

To  find  the  depth  on  the  weir,  corrected  for  the  leakage  of  the  wheelpit,  let 
h'  =  the  depth  on  the  weir  by  observation, 


80  EXPERIMENTS   ON  THE   FLOW  OF   WATER   OVER   WEIRS. 

h"  =  ihe  depth  on  the  weir  corrected  for  the  leakage, 
1=  length  of  the  weir, 
Q  =  the  quantity  passing  over  the  weir,  the  dimensions  being  all  in  feet. 


We   have    Q-\-L  =  ihe    total    quantity    entering    the    wheelpit,    and    which    would 
have   passed    over   the   weir,  if  there  had   been   no   leakage   out   of  the   wheelpit. 

To  determine  the  corrected  depth,  it  is  necessary  to  assume  some  formula 
giving  nearly  the  relations  between  the  quantities  h',  I,  and  Q.  Let  us  use  that 
given  by  Lesbros*  for  a  depth  of  0.20  metres  and  complete  contraction,  which, 
when  reduced  to  the  English  foot  as  the  unit,  and  adopting  our  own  notation,  is 


we  shall  have  also 

by  subtraction 

from  which  we  derive 


or  substituting  for   L  its  value    0.03112  \^,  we  have 

,„ (,&  .  0.03112^X3 

3.12/     /  ' 

By  this  formula,  the  reduced  heights  given  in  column  12  have  been  obtained. 

131.  COLUMN  15.     Fall  from  the   surface   of  water  in  the  forebay,  to  the  surface  of 
the   water  in    the    wheelpit.      This    is    obtained    by   taking    the    difference   of  the    cor- 

,  responding   numbers   in   columns   13    and    14. 

132.  COLUMN  16.     Uniform  fall  from  the  forebay  to  the  wheelpit,  to  which  the  depths 
on  the  tceir  in  each  series  are  reduced.      The  fall  in  the  same  series  given  in  column 
15,  which  is  the   nearest   to   the   mean   fall    in  all    the   experiments    in   the   series, 
is  assumed  for  this  purpose ;    it  is  unimportant  what  fall  is  taken,  provided  it   is 
near  the   mean. 

133.  COLUMN  17.     Depth  on  the  weir  corrected  for  the  leakage  of  the  wheelpit,  and 
the  variation  in  the  fall.      It  must    be  recollected   that   all   the   experiments  of  each 


*  Experiences  Hydrauliques  sur  les  lots  de  I'ecoulement  de   I'eau,  by  M.  Lesbros,  Paris :    1851.      Table 
XXXIX. 


EXPERIMENTS   ON  THE    FLOW   OF   WATER  OVER   WEIRS.  81 

series,  were  made  with  the  same  opening  of  the  regulating  gate  of  the  turbine ; 
that  is,  the  areas  of  the  orifices  through  which  the  water  entered  the  wheelpit, 
were  the  same  in  each.  In  all  the  experiments,  a  small  quantity  of  the  water 
entering  the  wheelpit,  passed  between  the  gate  and  the  lower  curb,  in  consequence 
of  the  leather  packing  not  being  perfectly  adjusted ;  this  did  not  affect  the  results, 
however,  as  these  orifices  were  also  submerged  in  the  wheelpit.  Under  these  cir- 
cumstances, if  the  head  had  been  constant,  the  quantity  of  water  entering  the 
wheelpit,  would  also  have  been  constant;  but  the  head  was  subject  to  a  varia- 
tion, comparatively  small  certainly,  but  sufficient  to  produce  a  material  change  in 
the  quantity  of  water  entering  the  wheelpit,  and,  consequently,  in  the  depth  on 
the  weir. 

To  clear  the  results  from  this  source  of  irregularity,  it  will  be  necessary  to 
ascertain  what  the  depths  on  the  weir  would  have  been,  if  the  head  had  been 
constant.  For  this  purpose,  let 

ff=  the  constant  head  to  which  the  depths  on  the  weir,  in  any  particular 
series,  are  to  be  reduced,  and  which  varies  but  little  from  the  actual 
heads  in  the  same  series; 

J7'— the    actual   head   in   the   particular   experiment   to   be   reduced; 
H"  —  the    depth    on    the   weir,   corrected    for   the   variation   of    the   head,   or 

corresponding   to   the    constant   head   H; 

A"  — the    depth   on   the   Aveir   corresponding    to   the   head    H',  and   which   is 
the    depth  given   by    observation,   corrected  for  the    leakage   of   the 
wheelpit ; 
<?  =  the    quantity    of    water,    in    cubic    feet    per    second,    that    would    have 

entered   the   wheelpit,  if  the   head   had   been   II; 
q'  =  the    quantity   of   water    corresponding    to    the    head   H',   and   which   is 

the   same   as   Q  -j-  L  (art.  130) ; 
^— the   length   of  the  weir; 

C—  the   coefficient   of  the   formula   for   the    discharge    over  weirs ; 
a,  a',  a",  etc.  =  the   areas    of    the    several    orifices    through    which    the   water    entered 
the   wheelpit,   all   of  them   being   submerged   in   the   wheelpit; 

c,c',c", etc.  =  the  corresponding   coefficients   of  contraction; 

v 

q'  =  c  a  ^2ffH'  -f-  c'a'  V  2gH'  4-  c  "a"  \]2gH'-{-  etc. ; 


/  =  (e  a  +  c'a'  -f  c"a"+  etc.)  V  2gH' ; 
11 


82  EXPERIMENTS   ON   THE   FLOW  OF  WATER  OVER   WEIRS 

similarly 

g 

by  division, 


also 

4=Clh"*  and  q= 

whence, 

i: 
r 

therefore, 


H    —hr"> 

whence,   we   derive 

•LHI  _  -uit  ff\^ 
h   —h    -,    . 


By  this  last  formula,  the  corrected  depths  given  in  column  17  have  been 
computed. 

By  an  inspection  of  column  13,  it  will  be  seen  that  the  level  of  the  water 
above  the  wheel  was  maintained  throughout  each  series  with  great  uniformity, 
excepting  in  a  few  experiments  in  which  it  was  intentionally  altered,  as  will  be 
seen  presently.  The  height  of  the  water  in  the  wheelpit  necessarily  varied  with 
the  depth  upon  the  weir,  and  this  is  the  principal  cause  of  the  variations  in 
the  fall. 

Several  of  the  experiments  given  in  table  X.,  were  made  for  the  express 
purpose  of  testing  the  accuracy  of  the  method  of  reduction  just  described.  Thus, 
in  experiments  41  and  42,  the  weir  was  in  the  same  state  as  in  experiment  40, 
but  the  height  of  the  water  above  the  wheel  was  lowered,  and  the  differences 
in  the  observed  depths  upon  the  weir,  given  in  column  9,  are  to  be  attributed 
entirely  to  the  diminution  in  the  quantity  of  water  entering  the  wheelpit,  in  con- 
sequence 6f  the  diminished  head.  If  the  method  of  reduction  is  accurate,  how- 
ever, the  corrected  depths  in  these  three  experiments,  given  in  column  17,  should 
be  the  same. 

In  table  VIII,  are  collected  all  the  experiments  made  for  this  object,  together 
with  the  other  experiments  forming  part  of  the  corresponding  series,  with  which 
they  may  be  compared,  the  weir  having  been  in  the  same  state. 


EXPERIMENTS   ON  THE   FLOW   OF   WATER  OVER   WEIRS. 


83 


TABLE  VIII. 


Number 
of  the 
experiment. 

Fall  from  the 
forebay  to  the 
wheelpit. 
Feet. 

Corrected  depth 
upon  the  weir,  in 
Feet. 

Variation  in  the  fall 
from  the  initial 
experiment. 
Feet. 

Variation  in  the  cor- 
rected depth,  from  the 
initial  experiment. 
Feet. 

40 

14.088 

0.79096 

41 

13.554 

0.79049 

—  0.534 

—  0.00047 

42 

13.149 

0.78976 

—  0.939 

—  0.00120 

49 

13.904 

0.95477 

52 

13.436 

0.95380 

—  0.468 

—  0.00097 

53 

12.962 

0.95097 

—  0.942 

—  0.00380' 

63 

13.719 

1.13177 

64 

12.806 

1.12508 

—  0.913 

—  0.00669 

72 

13.816 

0.92170 

73 

13.315 

0.92145 

—  0.501 

—  0.00025 

74 

,12.665 

0.92153 

—  1.151 

—  0.00017 

It  will  be  perceived  that  the  variations  in  the  fall,  to  which  the  method  ol 
reduction  is  applied  in  these  experiments,  are,  nearly  all  of  them,  much  greater 
than  any  that  occur  in  the  regular  experiments.  This  was  arranged  for  the  pur- 
pose of  applying  an  extreme  test  to  the  method.  Several  of  the  variations  in 
the  corrected  depths,  are  not  within  the  limits  of  ordinary  observation ;  several 
of  them,  however,  are  sensible,  and  being  all  in  the  same  direction,  they  cannot 
be  attributed  entirely  to  errors  of  observation,  but,  in  part  at  least,  to  either  a 
slight  defect  in  the  method  of  reduction,  or  to  the  instability  of  the  apparatus. 

It  was  observed  during  the  course  of  the  experiments,  that  the  quantity  of 
water  entering  the  wheelpit,  sometimes  diminished  sensibly,  although  no  change 
had  been  made  in  the  height  of  the  regulating  gate ;  the  precaution  having  been 
taken  to  fix,  in  a  secure  manner,  the  apparatus  by  which  the  gate  was  moved. 
At  the  time  the  experiments  were  made,  this  change  was  attributed  to  a  minute 
lowering  •  of  the  gate,  taking  place  very  slowly,  and  arising  from  a  defect  in  the 
stiffness  of  the  apparatus,  aided  by  a  slight,  but  not  totally  insensible  vibration 
of  the  whole  apparatus,  caused  by  the  passage  of  the  water  through  the  aper- 
tures. To  show  how  minute  a  change  in  the  height  of  the  regulating  gate, 
would  produce  the  observed  changes  in  the  quantity,  let  us  take  the  two  first 
experiments  given  in  table  IX.  The  regulating  gate  was  raised  to  a  height  not 


84 


EXPERIMENTS   ON  THE   FLOW   OF   WATER  OVER   WEIRS. 


exceeding  0.01  feet;  supposing  it  to  have  been  at  just  that  height,  and  that 
any  change  in  its  height  would  have  produced  an  equal  proportional  change  in 
the  discharge,  the  observed  proportional  change  in  the  quantity  was  0.00046 ; 
consequently,  the  absolute  change  in  the  height  of  the  gate  must  have  been 
0.0000046  feet. 

In  order  to  prevent  this  source  of  irregularity  from  affecting  the  experi- 
mentS,  the  regulating  gate  was  usually  set  some  hours  before  the  experiments 
were  made.  This  probably  obviated  the  difficulty  in  part,  but  not  entirely,  as 
will  be  seen  by  table  IX.,  in  which  are  collected  all  the  experiments  that  were 
repeated  under  identical  circumstances. 


TABLE    IX. 


Number 
of  the 
experiment. 

Number  of 
the  series. 

Corrected  depth  upon 
the  weir,  in  feet. 

Variation  in  the  depth 
from  the  initial  experi- 
ment, in  feet. 

Proportional  change  in 
the  quantities  that 
entered  the  wheelpit. 

Time  that  had  elapsed 
when  the  experiment 
was  made,  since  the 
gate  was  Bet. 

Hours. 

Minutes. 

3 

7 

I. 

a 

0.19583 
0.19577 

—  0.00006 

—  0.00046 

8 
12 

ii. 

a 

0.23386 
0.23505 

• 

4-  0.00119 

4-  0.00764 

0 

1 

'  22 
31 

16 
20 
24 

m. 

tt 
tt 

0.29223 
0.29166 
0.29210 

—  0.00057 
—  0.00013 

—  0.00292 
—  0.00067 

4 
5 
6 

33 
39 
39 

26 
30 

IV. 

u 

1.06532 
1.06548 

-f-0.00016 

+  0.00023 

16 

17 

58 
51 

35 
40 

V. 

u 

.0.79190 
0.79096 

—  0.00094 

—  0.00178 

2 
3 

25 
34 

44 
49 

VI. 

tl 

0.95656 
0.95477 

—  0.00179 

—  0.00281 

5 
6 

20 
40 

55 
58 
63 

VII. 

u 
u 

1.13356 
1.13306 
1.13177 

—  0.00050 
—  0.00179 

—  0.00066 
—  0.00237 

2 
3 

4 

26 
31 
39 

66 
69 

VIII. 

ft 

1.06358 
1.06272 

—  0.00086 

—  0.00121 

3 
3 

08 
46 

Mean  pr 

the  sig 

Dportional  change  in  the  quantity,  neglecting 

0.00208 

EXPERIMENTS  ON  THE  FLOW  OF   WATER  OVER  WEIRS.  85 

Although  the  variations  in  the  depths  given  in  the  preceding  table  are  very 
small,  the  fact  that  they  are  nearly  all  negative  precludes  the  idea  that  they  are 
entirely  due  to  errors  of  observation  ;  we  must,  therefore,  attribute  to  some  other 
cause  a  portion  of  the  irregularity. 

134.  COLUMN  19.  Combination  of  experiments  used  to  determine  the  value  of  a.  It 
has  been  shown  (art.  122)  how,  by  means  of  two  experiments  in  which  the 
quantities  passing  over  different  weirs  are  equal,  we  may  determine  a  in  the 
formula 

Q  =  Clh*. 

We   now  propose   to   show  how,  by  means   of  two   such   experiments,  the  value   of 
a  may  be   found   in   the   proposed   formula 

Q=  C(l—bnh}h\ 

In  this  equation,  we  have  I  and  a  constant  quantities  to  be  determined; 
we  have  also  C  a  constant,  which  we  may  here  consider  as  indeterminate;  the 
same  may  be  said  of  Q,  as  limited  to  the  experiments  in  the  same  series. 

Let  I,  n,  and  h,  represent  the  length  of  the  weir,  the  number  of  end  con- 
tractions, and  the  depth  upon  the  weir  in  one  experiment;  and  li,  ni}  and  hi,  the 
corresponding  quantities  in  another  experiment  of  the  same  series;  we  have 

Q=C(l—  bnh)ha; 
.  and 


since   for  the   same    series    Q  is   constant,  we   have 

(l—bnh}h*=(ll—bnlhl}hla: 
taking  the   logarithms, 

,  a  Log.  h  -\-  Log.  (I  —  bnh]  =  a  Log.  hi-\-  Log.  (^ 
whence   we   derive 

_  Log.  (/!  —  Jjij^j)  —  Log.  (I  —  bnfi) 
Log.  h  —  Log.  AI 

This   equation  is  still  indeterminate,  but  can  be   rendered   determinate,  by  assuming 
a   value   for  b. 

If  the  formula  represents  the  true  law,  and  the  experiments  from  which  the 
values  of  the  constants  are  to  be  derived  are  perfectly  accurate,  the  particular 
combination  of  experiments  to  be  used  is  evidently  unimportant.  As  such  an 


86  EXPERIMENTS  ON  THE   FLOW  OF  WATER  OVER  WEIRS. 

assumption  would  be  very  unreasonable,  we  have  combined  the  experiments,  with 
a  view  of  obtaining  the  best  approximation  from  imperfect  data ;  and  this  we 
have  accomplished  by  selecting  experiments  the  most  remote  from  each  other  in 
the  values  of  the  respective  data  they  furnish;  thus,  in  series  I.,  the  combina- 
tions are  made  by  combining  experiment  6,  in  which  I  has  the  least,  and,  con- 
sequently, h  the  greatest  value,  with  each  of  the  others,  omitting  entirely  all  the 
experiments  which,  for  any  reason,  appear  to  be  unsuitable. 

Generally,  in  each  series,  one  experiment  has  been  repeated  as  a  test,  in 
order  to  show  if  any  change  had  taken  place  in  the  apparatus;  thus,  in  series 
III.,  experiments  16,  20,  and  24,  were  made,  so  far  as  is  known,  under  identical 
circumstances ;  in  such  cases,  means  deduced  from  the  repeated  experiments  have 
been  used  instead  of  making  a  separate  combination  with  each. 

135.  COLUMNS  numbered  20  to  25.  Values  of  a  when  £  =  0.07,  I  =  0.065, 
etc.  The  object  is,  to  find  the  values  of  a  and  I,  in  the  formula 

Q=  C(l—lnh}h", 

that  will  give  to  the  computed  discharges  in  each  series  the  most  uniform  results. 
For  this  purpose,  successive  values  of  b  are  assumed,  and  the  corresponding  values 
of  a,  determined.  The  value  of  I  leading  to  values  of  a,  having  the  least  vari- 
ation among  themselves,  will  evidently  be  that  most  nearly  fulfilling  this  condition. 
To  aid  in  the  selection  of  the  proper  value  of  I,  the  table  gives  the  differences 
between  the  values  of  a  deduced  from  each  combination,  and  the  mean  value  of 
a  deduced  from  all  the  combinations,  with  the  same  value  of  b,  and  the  sums  of 
these  differences  (having  no  regard  to  the  sign)  are  also  given.  If  will  be  seen 
that  the  sum  of  the  differences  is  least  when  the  value  of  b  =  0.05,  the  corre- 
sponding mean  value  of  a  being  1.46994,  or  1.47  very  nearly ;  consequently,  to 
represent  the  whole  of  the  experiments  with  the  most  uniformity,  the  formula 
becomes 

Q=C(l— 


88 


EXPERIMENTS   ON  THE   FLOW   OF   WATER  OVER  WEIRS. 


TABLE    X. 

EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS,  MADE  AT  THE  TREMONT  TURBINE. 


1 

3 

3 

4 

5 

6 

7 

8 

9 

1O 

11 

Temperature  of 

the  atmosphere 

TIME. 

No 

Height 

in  degrees  of 

Dura- 

Total 

of 

Reference 

Depth  of     of  the 

Fall 

affect- 

Number of  the 
series  and  of 
the  experiment. 

Date  of  the  experiment 
1861. 

thermometer. 

Beginning 
of  the 
experiment. 

Ending 
of  the 
experiment. 

tion 
of  the 
experi- 
ment, 
n  min- 

length of 
the  weir, 
in  feet. 
I 

the 
end 
con- 
trac- 
tions. 

to  the 
figures  on 
plate  X. 

water  on 
the  weir 
by  obser- 
vation ; 
In  feet. 

In  the 
lower 
canal, 
below 
the  top 
of  the 

ing  the 
leakage 
of  the 
wheel- 
pit;  in 
feet. 

External 
ilr  in  the 

Near  the 

•weir. 

utes. 

hi 

weir, 

A. 

shade. 

H. 

inin. 

H. 

tiiin. 

in  feet. 

Series  I. 

Exp.  1 

January  30,  A.M. 

10 

0 

10 

12 

12 

6.987 

2 

Fig.  1 

0.3125 

1.16 

1.47 

«     2 

II               11            II 

10 

18 

10 

25 

7 

13.978 

4 

«     2    0.1948 

1.16 

1.35 

«     3 

11               II            II 

6.50 

10 

39 

10 

46.5 

7.5 

13.978 

8 

"     3    0.1952 

1.16 

1.36 

«     4 

II          II        << 

6.25 

31.50 

11 

2 

11 

6 

4 

10.482 

6 

"     4   0.2389 

1.16 

1.40 

«     5 

II               II           II 

5.75 

31.00 

11 

20 

11 

26 

6 

7.000 

4 

«     51  0.3149 

1.17 

1.48 

«      6 

a           u        a 

6.25 

11 

40 

11 

45 

5 

3.500 

2 

«     6 

0.5028 

1.25 

1.75 

"     7 

u           u        a 

5.75 

11 

52 

11 

55 

3 

13.978 

8 

«    3 

0.1951 

1.22 

1.42 

Series  II. 

Exp.  8 

January  30,  P.M. 

30.75 

2 

22 

2 

26 

4 

13.978 

8 

Fig.  3 

0.2330 

1.10 

1.33 

"      9 

u           u        u 

4.50 

30.50 

2 

35.5 

2 

41 

5.5 

10.482 

6 

ii    4 

0.2842 

1.10 

1.38 

"    10 

u          u        u 

4.50 

30.50 

2 

54 

3 

0 

6 

7.000 

4 

«    5 

0.3738 

1.10 

1.47 

«    11 

II               II           (( 

4.25 

3 

15 

3 

21 

6 

3.500 

2 

«     6 

0.5973 

1.20 

1.80 

"    12 

II          ti        II 

4.00 

30.75 

3 

31 

3 

38 

7 

13.978 

8 

«    3 

0.2341 

1.27 

1.50 

«    13 

11               U           U 

3.50 

30.50 

3 

53 

3 

59 

6 

13.978 

4 

"    2 

0.2330 

1.22 

1.45 

«    14 

u          u        u 

4 

11 

4 

16.5 

5.5 

6.987 

2 

«     1 

0.3719 

1.10 

1.47 

«    15 

II            11         11 

2.75 

30.75 

4 

24 

4 

32 

8 

16.980 

4 

"    7 

0.2046 

1.09 

1.29 

Series  III. 

Exp.  16 

January  31,  P.M. 

5.00 

31.00 

2 

23.5 

2 

32 

8.5 

13.978 

4 

Fig.  2 

0.2916 

1.17 

1.46 

"    17 

11         11       11 

5.00 

31.25 

2 

41 

2 

49.5 

8.5 

6.987 

2 

"     1 

0.4652 

1.17 

1.64 

«    18 

U               1             11 

5.00 

31.25 

2 

56.5 

3 

5 

8.5 

13.978 

8 

"    3 

0.2932 

1.16 

1.45 

"    19 

11            1          11 

5.00 

31.00 

3 

12 

3 

18 

6 

10.484 

6 

11     4 

0.3564 

1.13 

1.49 

«    20 

11            1          11 

5.00 

30.75 

3 

29.5 

3 

35.5 

6 

13.978 

4 

"    2 

0.2910 

1.12 

1.41 

«    21 

1C               1             11 

4.50 

30.50 

3 

46 

3 

53 

7 

6.989 

4 

«    5 

0.4684 

1.15 

1.62 

«    22 

U               1             1C 

4.50 

4 

2.5 

4 

8.5 

6 

3.500 

2 

"    8 

0.7478 

1.12 

1.87 

«   23 

11          II        11 

4.25 

31.00 

4 

14 

4 

20 

6 

16.980 

4 

11    7 

0.2548 

1.12 

1.37 

"   24 

II          11        II 

4.00 

4 

29.5 

4 

35.5 

6 

13.978 

4 

"    2 

0.2914 

1.16 

1.45 

Series  IV. 

Exp.  25 

February  1,  A.M. 

5.00 

31.00 

9 

15 

9 

21 

6 

13.978 

4 

Fig.  2 

0.4071 

1.14 

1.55 

"    26 

ii         ti     s  ii 

7.00 

9 

38.5 

9 

46 

7.5 

3.496 

2 

"    8 

1.0447 

1.12 

2.16 

«    27 

u         u       it 

8.50 

30.00 

9 

52 

9 

57 

5 

6.989 

4 

"    5 

0.6577 

1.15 

1.81 

«    28 

II               1C             1 

10.00 

10 

4 

10 

11 

7 

10.484 

6 

11    4 

0.4977 

1.10 

1.60 

«    29 

II            II          1 

10.00 

30.50 

10 

16 

10 

21 

5 

13.978 

8 

"     3 

0.4096 

1.15 

1.56 

«    30 

II            II          1 

10.50 

10 

31.5 

10 

39.5 

8 

3.496 

2 

«    8 

1.0456 

1.17 

2.22 

«   31 

II          II         I 

14.25 

31.50 

10 

57 

11 

1 

4 

3.496 

2 

"    8 

1.0452 

1.12 

2.17 

«    32 

II          11         1 

15.00 

11 

19 

11 

24.5 

5.5 

6.987 

2 

"     1 

0.6494 

1.17 

1.82 

«    33 

II          11        II 

15.50 

30.75 

11 

29.5 

11 

36 

6.5 

16.980 

4 

"    7 

0.3576 

1.22 

1.58 

EXPERIMENTS   ON   THE   FLOW   OF   WATER   OVER   WEIRS. 


89 


TABLE     X  —  CONTINUED. 
EXPERIMENTS   ON  THE  FLOW  OF  WATER  OVER  WEIRS,  MADE  AT  THE  TREMONT  TURBINE. 


13 

13 

14 

15 

16 

17 

18 

Number  of  the 
scries  and  of 
the  experiment. 

Depth  of  water 
on  the  weir, 
corrected  for 
the  leakage 
of  the 
wheelpit,  in 

Height  of 

water 
above  the 
wheel, 
taken  in  the 
forcbay  j 

Height  of 
the  water 
in  the 
wheelpit  ; 

Fall  from  the 
surface  of  the 
water  in  the 
forcbay,  to 
the  surface 
of  the  water 
in  the 
wheelpit  ; 

Uniform  fall 
from  the 
forehay  to 
the  wheelpit, 
to  which  the 
depths  on  the 
weir  in  each 
series  are 

Depth  on  the 

weir,  corrected 
for  the  leakage 
of  the  wheelpit, 
and  the  varia- 
tion  in  the  fall  ; 

REMARKS. 

feet. 
A//. 

in  feet. 

infect. 

in  feet. 

a'. 

reduced  ; 
in  feet. 
H, 

in  feet. 
A'". 

Series  I. 

Exp.  1 

0.31456 

14.869 

0.320 

14.549 

14.549 

0.31456 

In  experiments  2,  3,  and  7,  the  contrac- 

"      2 

0.19605 

14.896 

0.201 

14.695 

ft 

0.19540 

tion  was  incomplete,  as  the  water  fol- 

«      3 

0.19645 

14.894 

0.205 

14.689 

It 

0.19583 

lowed  the  top  of  the  weir. 

a       4 

0.24043 

14.881 

0.247 

14.634 

11 

0.23997 

«       5 

0.31696 

14.892 

0.320 

14.572 

It 

0.31679 

«       6 

0.50634 

14.876 

0.510 

14.366 

tt 

0.50848 

"      7 

0.19638 

14.886 

0.200 

14.686 

tt 

0.19577 

Series  II. 

Exp.  8 

0.23414 

14.910 

0.240 

14.670 

14.619 

0.23386 

In  experiment  15  the  contraction  was 

«       9 

0.28560 

14.909 

0.290 

14.619 

C4 

0.285CO 

incomplete,  as  the  water  followed  the  top 

«     10 

0.37568 

14.915 

0.380 

14.535 

u 

0.37640 

of  the  weir. 

"     11 

0.60059 

14.916 

0.610 

14.306 

It 

0.60494 

"     12 

0.23530 

14.906  !     0.240 

14.666 

u 

0.23505 

«     13 

0.23419 

14.912 

0.240 

14.672 

u 

0.23390 

«     14 

vO.37379 

14.910 

0.380 

14.530 

(( 

0.37455 

«     15 

0.20558 

14.918 

0.210 

14.708 

11 

0.20517 

Series  III. 

Exp.  16 

0.29266 

14.897 

0.300 

14.597 

14.532 

0.29223 

"     17 

0.46698 

14.900 

0.470 

14.430 

tt 

0.46808 

«     18 

0.29426 

14.897 

0.300 

14.597 

(i 

0.29382 

«     19 

0.35770 

14.897 

0.365 

14.532 

« 

0.35770 

«    20 

0.29205 

14.890 

0.300 

14.590 

« 

0.29166 

"     21 

0.47017  |     14.887 

0.477 

14.410 

u 

0.47149 

«    22 

0.75080 

14.883 

0.760 

14.123 

u 

0.75798 

«     23 

0.25571 

14.878 

0.260 

14.618 

cc 

0.25520 

"     24 

0.29246 

14.886 

0.300 

14.586 

• 

0.29210 

Series  IV. 

Exp.  25 

"     26 
«     27 
«     28 

0.40803 
1.04743 
0.65928 
0.49884 

14.904 
14.877 
14.871 
14.877 

0.420 
1.060 
0.670 
0.510 

14.484 
13.817 
14.201 
14.367 

14.537 

it 
tt 
tt 

0.40852 
1.06532 
0.66444 
0.50080 

In  experiments  26  and  30,  the  water 
flowing  over  the  weir  fell  upon  a  board 
placed  upon  the  brackets  N,  figures  1  and 
2,  plate  V.  ;    in  experiment  31   the  board 
was  removed.     So  far  as  is  known  the 

«     29       0.41053       14.893       0.420       14.473 
"    30       1.04837       14.908       1.060       13.848' 
"     31        1.04794  ;     14.886       1.060       13.826 

tt 
tf 
tt 

0.41113 
1.06548 
1.06560 

three  experiments   were  identical  in  all  ; 
other  respects.    By  comparing  the  cor- 
rected depths  upon  the  weir  given  in  col- 

"   32       0.65099       14.904       0.660 
"    33       0.35842  :     14.907       0.370 

14.244 
14.537 

(I 
tt 

0.65543 
0.35842 

umn  1  7,  it  appears  that  the  board  offered  no 
appreciable  obstruction  to  the  discharge. 

12 


90 


EXPERIMENTS   ON  THE   FLOW   OF    WATER   OVER   WEIRS. 


TABLE     X—  CONTINUED. 
EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS,  MADE  AT  THE  TREMONT  TUBBING. 


19 

20 

31 

33 

6  =  0.07 

6  =  0.065. 

6  =  0.06. 

Number  of  the 

series  and 
of  the 
experiment. 

Combination  of  experi- 
ments used  to  determine 
the  value  of  a. 

Values 

Differences  from  the 
mean  value  of  a,  or  from 
1.47797. 

Values 

Differences  from  the 
mean  value  of  a,  or  from 
1.47695. 

Values 

Differences  from  the 
mean  value  of  a,  or  from 
1.47394. 

of  a. 

of  a. 

of  a. 

+ 

- 

+ 

- 

+ 

- 

Series  I. 

Exp.  1 

1  and  6 

1.4691 

0.00887 

1.4669 

0.00905 

1.4648 

0.00914 

«     2 

«     3 

"     4 

4    «    6 

1.4753 

0.00267 

1.4742 

0.00175 

1.4731 

0.00084 

«     5 

5    "    6 

1.4814 

0.00343 

1.4801 

0.00415 

1.4789 

0.00496 

«     6 

"     7 

Series  II. 

Exp.  8 

8,  12,  and  11 

1.4768 

0.00117 

1.4756 

0.00035 

1.4744 

0.00046 

"      9 

9    "    11 

1.4787 

0.00073 

1.4775 

0.00155 

1.4763 

0.00236 

«    10 

10    "    11 

1.4805 

0.00253 

1.4791 

0.00315 

1.4777 

0.00376 

"    11 

"    12 

8,  12,    "    11 

1.4768 

0.00117 

1.4756 

0.00035 

1.4744 

0.00046 

«    13 

13    "    11 

1.4781 

0.00013 

1.4766 

0.00065 

1.4751 

0.00116 

"    14 

14    «    11 

1.4774 

0.00057 

1.4748 

0.00115 

1.4723 

0.00164 

"    15 

15    "    11 

1.4800 

0.00203 

1.4786 

0.00265 

1.4772 

0.00326 

Series  III. 

, 

Exp.  16 

16,20,  24,  and  22 

1.4778 

0.00017 

1.4759 

0.00005 

1.4740 

0.00006 

"    17 

17  "    22 

1.4784 

0.00043 

1.4752 

0.00075 

1.4721 

0.00184 

"    18 

18  «    22 

1.4811 

0.00313 

1.4797 

0.00375 

1.4782 

0.00426 

"    19 

19  "    22 

1.4827 

0.00473 

1.4811 

0.00515 

1.4795 

0.00556 

«    20 

16,20,24,  "    22 

1.4778 

0.00017 

1.4759 

0.00005 

1.4740 

0.00006 

"    21 

21  "    22 

1.4814 

0.00343 

1.4796 

0.00365 

1.4778 

0.00386 

«   22 

«    23 

23  «    22 

1.4752 

0.00277 

1.4734 

0.00255 

1.4716 

0.00234 

"   24 

16,  20,  24,  "    22 

1.4778 

0.00017 

1.4759 

0.00005 

1.4740 

0.00006 

Series  IV. 

Exp.  25 

25  and  26,  30 

1.4827 

0.00473 

1.4800 

0.00405 

1.4773 

0.00336 

"   26 

«   27 

27    "    26,30 

1.5023 

0.02433 

1.4997 

0.02375 

1.4971 

0.02316 

«   28 

28    "    26,  30 

1.4857 

0.00773 

1.4834 

0.00745 

1.4812 

0.00726 

«   29 

29    "    26,  30 

1.4838 

0.00583 

1.4817 

0.00575 

1.4796 

0.00566 

«   30 

«   81 

"   32 

32    "    26,30 

1.4878 

0.00983 

1.4832 

0.00725 

1.4786 

0.00466 

«   33 

33    "    26,30 

1.4853 

0.00733 

1.4828 

0.00685 

1.4802 

0.00626 

EXPERIMENTS   ON  THE   FLOW   OF   WATER   OVER   WEIRS. 


91 


TABLE     X  —  CONTINUED. 
EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS,  MADE  AT  THE  TREMONT  TURBINE. 


33 

24 

35 

6-0.055 

6  =  0.05. 

6  =  0.045. 

Number  of  the 

series  and 
of  the 

Differences  from  the 

Differences  from  the 

Differences  from  the 

experiment. 

Values 

mean  value  of  a,  or  from 
1.47194. 

Values 

mean  value  of  a,  or  from 
1.46994. 

Values 

mean  value  of  a,  or  from 
1.46795. 

of  a. 

of  a. 

of  a. 

+ 

- 

+ 

- 

+ 

- 

Series  I. 

Exp.    1 

1.4626 

0.00934 

1.4605 

0.00944 

1.4584 

0.00955 

«       2 

«       3 

«       4 

1.4721 

0.00016 

1.4710 

0.00106 

1.4700 

0.00205 

"       5 

1.4778 

0.00586 

1.4765 

0.00656 

1.4754 

0.00745 

«       6 

a         7 

Series  H. 

Exp.   8 

1.4733 

0.00136 

1.4722 

0.00226 

1.4710 

0.00305 

"       9 

1.4750 

0.00306 

1.4737 

0.00376 

1.4725 

0.00455 

V 

«     10 

1.4762 

0.00426 

1.4749 

0.00496 

1.4734 

0.00545 

«     11 

«     12 

1.4733 

0.00136 

1.4722 

0.00226 

1.4710 

0.00305 

«     13 

1.4735 

0.00156 

1.4721 

0.00216 

1.4706 

0.00265 

a       14 

1.4697 

0.00224 

1.4671 

0.00284 

1.4646 

0.00335 

«     15 

1.4758 

0.00386 

1.4744 

0.00446 

1.4730 

0.00505 

Series  III. 

Exp.   16 

1.4721 

0.00016 

1.4702 

0.00026 

1.4683 

0.00035 

"     17 

1.4688 

0.00314 

1.4656 

0.00434 

1.4624 

0.00555 

"     18 

1.4768 

0.00486 

1.4753 

0.00536 

1.4739 

0.00595 

«     19 

1.4780 

0.00606 

1.4764 

0.00646 

1.4748 

0.00685 

«     20 

1.4721 

0.00016 

1.4702 

0.00026 

1.4683 

0.00035 

"     21 

1.4760 

0.00406 

1.4742 

0.00426 

1.4725 

0.00455 

«     22 

«     23 

1.4699 

0.00204 

1.4681 

0.00184 

1.4663 

0.00165 

"     24 

1.4721 

0.00016 

1.4702 

0.00026 

1.4683 

0.00035 

Series  IV. 

Exp.   25 

1.4746 

0.00266 

1.4720 

0.00206 

1.4693 

0.00135 

"     26 

«     27 

1.4945 

0.02256 

1.4920 

0.02206 

1.4895 

0.02155 

"     28 

1.4789 

0.00696 

1.4767 

0.00676 

1.4745 

0.00655 

"     29 

1.4776 

0.00566 

1.4755 

0.00556 

1.4735 

0.00555 

"     30 

"     31 

"     32 

1.4741 

0.00216 

1.4695 

0.00044 

1.4651 

0.00285 

"     33 

1.4777 

0.00576 

1.4752 

0.00526 

1.4728 

0.00485 

92 


EXPERIMENTS   ON  THE   FLOW   OF   WATER   OVER   WEIRS. 


TABLE     X  —  CONTINUED. 
EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS,  MADE  AT  THE  TREMONT  TURBINE. 


1 

a 

3 

4 

5 

6 

*y 

8 

9 

10 

11 

Temperature  of 
the  atmosphere 

TIME. 

Dura- 

Total 

No. 
of 

Depth  of 

Heij-ht 
of  the 

Fall 

Number  of  the 
series  and  of 
the  experiment. 

Date  of  the  experiment, 
1851. 

in  degrees  of 
Fahrenheit's 
thermometer. 

tion 
of  the 
experi- 
ment, 
in  min- 

length of 
the  weir, 
in  feet. 

;. 

the 
end 
con- 
trac- 
tions. 

Reference 
to  the 
figures  on 
plate  X. 

water  on 
the  weir 
by  obser- 
vation ; 
in  feet. 

water 
in  the 
lower 
canal, 
below 
the  top 
of  the 

affect- 
ing the 
leakage 
of  the 
wheel- 
pit;  in 
feet. 

Beginning 
of  the 
experiment. 

Ending 
of  the 
experiment. 

External 

iti  r  in  thi1 

Near  the 

shade. 

weir. 

H. 

mln. 

n. 

min. 

utes. 

n. 

h'. 

weir, 
in  feet 

h. 

Series  V. 

Exp.  34 

February  I,P.M. 

20.75 

31.50 

2 

11 

2 

15.5 

4.5 

13.978 

4 

Fig.  2 

0.4937 

1.12 

1.61 

"     35 

it           it        a 

20.00 

31.50 

2 

25 

2 

33 

8 

6.987 

2 

"     1 

0.7908 

1.16 

1.95 

"     36 

ti           it        it 

21.50 

31.50 

2 

39 

2 

43 

4 

13.978 

8 

"     3 

0.4981 

1.17 

1.67 

"     37 

a           it        it 

22.25 

31.50 

2 

49.5 

2 

54 

4.5 

10.484 

6 

u     4 

0.60CO 

1.23 

1.84 

"     38 

tl                 it             it 

20.50 

3 

3.5 

3 

14 

10.5 

6.989 

4 

"     5 

0.8000 

1.21 

2.01 

«     39 

it           u        u 

21.50 

30.00 

3 

19 

3 

23.5 

4.5 

16.980 

4 

"    7 

0.4337 

1.22 

1.65 

«      40 

a           a         a 

21.00 

31.50 

3 

34 

3 

48 

14 

6.987 

2 

"     1 

0.7896 

1.24 

2.03 

«      41 

a           a        « 

18.50 

31.25 

4 

16 

4 

26 

10 

6.987 

2 

"     1 

0.7790 

1.25 

2.03 

«      42 

u           u         u 

18.00 

4 

52 

5 

0 

8 

6.987 

2 

«     1 

0.7704 

1.35 

2.12 

Series  VI. 

Exp.  43 

February  3,  P.  M. 

38.25 

2 

6 

2 

11 

5 

13.978 

4 

Fig.  2 

0.5977 

1.10 

1.70 

"      44 

«         it       tt 

38.25 

32.25 

2 

20.5 

2 

30 

9.5 

6.987 

2 

«     1 

0.9561 

1.17 

2.13 

"      45 

a         tt       tt 

38.25 

2 

37 

2 

43 

6 

6.987 

4 

"     9 

0.9636 

1.17 

2.13 

«     46 

u           tt        tt 

37.50 

32.00 

2 

48.5 

2 

56 

7.5 

13.978 

8 

«    3 

0.6023 

1.16 

1.76 

«     47 

It          tt        tt 

37.50 

3 

6.5 

3 

13 

6.5 

10.488 

6 

"     4 

0.7308 

1.11 

1.84 

"     48 

tt            tt         tt 

37.25 

32.00 

3 

16.5 

3 

22.5 

6 

16.980 

4 

"    7 

0.5238 

1.13 

1.65 

"     49 

tl          It        tt 

37.00 

3 

40 

3 

45 

5 

6.987 

2 

«     1 

0.9533 

1.13 

2.08 

"     50 

tt           u        tt 

3 

47 

3 

59 

12 

6.987 

2 

"     1 

0.9531 

1.13 

2.08 

"     51 

tt            tt         tt 

4 

2 

4 

4 

2 

6.987 

2 

«     1 

0.9539 

1.13 

2.08 

"     52 

tt            tt         tl 

4 

23 

4 

31 

8 

6.987 

2 

"     1 

0.9415 

1.13 

2.07 

"     53 

tt            tl         11 

33.75 

4 

46.5 

5 

0 

13.5 

6.987 

2 

"     1 

0.9275 

1.14 

2.07 

Series  VII. 

Exp.  54 

February  4,  A.M. 

26.00 

31.75 

9 

6 

9 

12.5 

6.5 

16.980 

4 

Fig.  7 

0.5233 

1.12 

1.64 

«     55 

it         ti       tt 

26.50 

9 

26.5 

9 

37 

10.5 

5.487 

2 

"  10 

1.1278 

1.14 

2.27 

"     66 

it         tt       ti 

31.00 

31.75 

9 

44 

9 

57 

13 

6.987 

2 

"  11 

0.9544 

1.15 

2.10 

«     57 

ti         it       u 

30.00 

31.75 

10 

17 

10 

22 

5 

8.489 

2 

"  12 

0.8375 

1.14 

1.98 

"     58 

it         it       u 

31.25 

31.75 

10 

31 

10 

36 

5 

5.487 

2 

"  10 

1.1269 

1.17 

2.30 

"     59 

ti         ti       n 

33.50 

31.75 

10 

42 

10 

46.5 

4.5 

6.987 

4 

"  13 

0.9609 

1.13 

2.09 

"      60 

ti         it       ti 

34.00 

31.75 

10 

56.5 

11 

1 

4.5 

13.978 

8 

"    3 

0.6017 

1.13 

1.73 

"      61 

11                 11             It 

35.00 

11 

10 

11 

15 

5 

10.489 

6 

"    4 

0.7303 

1.12 

1.85 

«      62 

11            tl         11 

37.50 

31.75 

11 

20 

11 

26 

6 

13.978 

4 

"    2 

0.5971 

1.15 

1.75 

"      63 

11            It         tl 

38.00 

31.75 

11 

39.5 

11 

55 

15.5 

5.487 

2 

"  10 

1.1256 

1.14 

2.27 

"      64 

"            "    P.M. 

40.75 

0 

16 

0 

25 

9 

5.487 

2 

"  10 

1.0935 

1.13 

2.22 

Series  VIII. 

Exp.  65 

February  4,  P.M. 

38.25 

32.00 

3 

10.5 

3 

14 

3.5 

16.980 

4 

Fig.  7 

0.2316 

1.19 

1.42 

«      66 

It                 U             tl 

37.50 

3 

33.5 

3 

40 

6.5 

1.829 

2 

«  14 

1.0581 

1.17 

2.23 

"      67 

11                11             U 

37.00 

3 

45 

3 

52 

7 

3.658 

4 

"  15 

0.6650 

1.18 

1.84 

«      68 

tl          tl        11 

36.75 

3 

58 

4 

2 

4 

5.487 

6 

«  16 

0.5066 

1.24 

1.75 

"      69 

It            11         tl 

4 

11.5 

4 

16 

4.5 

1.829 

2 

"  14 

1.0574 

1.21 

2.27 

"     70 

11            It         It 

36.25 

32.00 

4 

21 

4 

25 

4 

8.489 

2 

"  12 

0.3706 

1.19 

1.56 

"     71 

tl                 tl             U 

4 

32 

4 

37 

5 

5.487 

2 

«  10 

0.4980 

1.19 

1.09 

Series  IX. 

Exp.  72 

February  4,  P.M. 

34.25 

4 

53 

5 

2 

9 

16.980 

4 

Fig.  7 

0.9206 

1.18 

2.10 

"      73 

it         ti      u 

34.25 

5 

17.5 

5 

24 

6.5 

16.980 

4 

"    7 

0.9091 

1.02 

1.93 

«      74 

It                 It             11 

33.75J 

5 

31 

5 

43 

12 

16.980 

4 

"    7 

0.8941]  1.17 

2.06 

EXPERIMENTS   ON   THE   FLOW   OF    WATER   OVER   WEIRS. 


93 


TABLE     X  —  CONTINUED. 
EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS,  MADE  AT  THE  TREMONT  TURBINE. 


12 

13 

14 

15 

16              17 

18 

Number  of  the 

Depth  of  water 
on  the  weir, 

Height  of 

water 

Height  of 

Fall  from  the 
surface  of  the 
water  in  the 

Uniform  fall 
from  the 

Depth  on  the 
weir,  corrected 

corrected  for 

above  the 

water 

forebay,  to 

the  wheelpit, 

lor  tne  leakage 

series  and  of 
the  experiment. 

the  leakage 
of  the 

wheel, 
taken  in  the 

in  the 

the  surface 
of  the  water 

to  which  the 
depths  on  the 
weir  in  each 

of  the  wheelpit, 
and  the  varia- 

REMARKS. 

wheelpit,  in 

forebay  ; 

wheelpit  ; 

in  the 
wheelpit  ; 

series  are 

tion  in  the  fall  ; 

feet. 
A". 

in  feet. 

infect. 

in  feet. 
IF. 

reduced  ; 
in  feet. 
H. 

in  feet. 
U". 

Series  V. 

Exp.  34 

0.49456 

14.845 

0.510 

14.335 

14.079 

0.49160 

In  experiments  41  and  42  the  weir  was 

"       35 

0.79229 

14.910 

0.810 

14.100 

tt 

0.79190 

in  the  same   state  as   in  experiment  40  ; 

"      36 

0.49897 

14.891 

0.520 

14.371 

« 

0.49557 

the  height  of  the  water  above  the  wheel 

"      37 

0.60710 

14.891 

0.620 

14.271 

tt 

0.60437 

was  reduced  for  the  purpose   of  testing 

"      38 

0.80151 

14.899 

0.820 

14.079 

tt 

0.80151 

the  method  of  reduction. 

"       39 

0.43446 

14.908 

0.450 

14.458 

tt 

0.43063 

"      40 

0.79112 

14.897 

0.809 

14.088 

tt 

0.79096 

"       41 

0.78054 

14.352 

0.798 

13.554 

U 

0.79049 

"      42 

0.77198 

13.939 

0.790 

13.149 

tt 

0.78976 

G(»l>Jpa  Y"  T 
Ot^l  lea     ¥  A. 

Exp.  43 

"       44 
«      45 
"       46 
"      47 

0.59850 
0.95752 
0.96501 
0.60311 
0.73181 

14.903 
14.929 
14.915 
14.894 
14.887 

0.620 
0.980 
0.992 
0.630 
0.755 

14.283 
13.949 
13.923 
14.264 
14.132 

13.907 
u 

tt 
tl 
tt 

0.59320 
0.95656 
0.96464 
0.59804 

0.72790 

In  experiments  50  and  51  the  weir  was  in 
the  same  state  as  in  experiment  49,  except- 
ing that  in  50  a  board  was  placed  on  the 
brackets  N,  figs.  1  and  2,  plate  V.,  on  which 
the  water  fell  ;  and  in  exp.  51,  the  plank  0, 
fig.  2,  plate  V.,  was  placed  in  the  position 

«      48 

0.52449 

14.889 

0.550 

14.339 

tl 

0.51917 

represented:  the  top  of  the  plank  was  6.5 

"       49 

0.95471 

14.884 

0.980 

13.904 

u 

0.95477 

inches  below  the  top  of  the  weir.     In  exps. 

«       50 

0.95451 

14.886 

0.980 

13.906 

It 

0.95453 

52  and  53,  the  weir  was  in  the  same  state  as 

"      51 

0.95530 

14.887 

0.980 

13.907 

tl 

0.95530 

in  exp.  49  ;    the  height  of  the  water  above 

«      52 

0.94291 

14.406 

0.970 

13.436 

tt 

0.95380 

the  wheel  was  lowered  for  the  purpose  of 

«      53 

0.92892 

13.914 

0.952 

12.962 

U 

0.95097 

testing  the  method  of  reduction. 

Series  VII. 

Exp.  54 

0.52399 

14.889 

0.550 

14.339 

13.882 

0.51837 

In  experiments  63  and  64  the  weir  was 

"       55 

1.12952 

14.884 

1.150 

13.734 

« 

1.13356 

in  the  same  state  ;   in  experiment  64  the 

"       56 

0.95581 

14.882 

0.980 

13.902 

a 

0.95535 

height  of  the  water  above  the  wheel  was 

"      57 

0.83870 

14.875 

0.865 

14.010 

tt 

0.83614 

lowered  for  the  purpose  of  testing  the 

"      58 

1.12863 

14.872 

1.152 

13.720 

tt 

1.13306 

method  of  reduction. 

"      59 

0.96230 

14.872 

0.990 

13.882 

tt 

0.96230 

"       60 

0.60251 

14.868 

0.629 

14.239 

it 

0.59743 

"       61 

0.73131 

14.865 

0.754 

14.111 

it 

0.72733 

"       62 

0.59791 

14.860 

0.620 

14.240 

tt 

0.59286 

"       63 

1.12732 

14.869 

1.150 

13.719 

it 

1.13177 

"       64 

1.09523 

13.926 

1.120 

12.806 

it 

1.12508 

Series  VIII. 

Exp.  65 

0.23257 

14.894 

0.240 

14.654 

13.839 

0.22817 

In  experiments  66,  67,  68,  and  69,  the 

"       66 

1.06337 

14.899 

1.068 

13.831 

U 

1.06358 

lengths  of  the  several  bays  of  the  weir 

"       67 

0.66802 

14.895 

0.676 

14.219 

11 

0.66202 

were  deemed  to  be  too  short  relative  to 

"       68 

0.50885 

14.902 

0.515 

14.387 

11 

0.50231 

the  depth  flowing  over,  for  the  proposed 

"       69 

1.06272 

14.905 

1.066 

13.839 

U 

1.06272 

formula  to  apply. 

"       70 

0.37221 

14.905 

0.380 

14.525 

11 

0.36625 

«       71 

0.50023 

14.909 

0.505 

14.404 

It 

0.49360 

Series  IX. 

72 

0.92119 

14.864 

1.048 

13.816 

13.839 

0.92170 

Experiments  72,  73,  and  74  were  made 

"      73 

0.90967 

14.350 

1.035 

13.315 

" 

0.92145 

for  the  express   purpose  of  testing  the 

"       74 

0.89469 

13.678 

1.013 

12.665 

u 

0.92153 

mode  of  reduction. 

94 


EXPERIMENTS   ON  THE   FLOW  OF    WATER   OVER   WEIRS. 


TABLE     X  —  CONTINUED. 
EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS,  MADE  AT  THE  TREMONT  TURBINE. 


19 

2O 

21 

22 

i  =  0.07 

b  =  0.066. 

6  =  0.08. 

Number  of  the 

Combination  of  experi- 

series and  of 

ments  used  to  determine 

Differences  from  the 

Differences  from  the 

Differences  from  the 

the  experiment 

the  yaluc  of  a. 

Values 

mean  value  of  a,  or  from 

Values 

mean  value  of  a.  or  from 

Values 

mean  value  of  a,  or  from 

1.47797. 

1.47595. 

1.47394. 

of  a. 

of  a. 

of  a. 

+ 

- 

+ 

- 

+ 

•     - 

Series  V. 

Exp.  34 

34  and  38 

1.4645 

0.01347 

1.4611 

0.01485 

1.4577 

0.01624 

"      35 

35,40,   "    39 

1.4737 

0.00427 

1.4726 

0.00335 

1.4716 

0.00234 

«     36 

36    "    38 

1.4679 

0.01007 

1.4660 

0.00995 

1.4G41 

0.00984 

"     37 

37    «    38 

1.4651 

, 

0.01287 

1.4630 

0.01295 

1.4610 

0.01294 

"     38 

«     39 

39    "    38 

1.4700 

0.00797 

1.4670 

0.00895 

1.4640 

0.00994 

«     40 

35,  40,   "    39 

1.4737 

0.00427 

1.4726 

0.00335 

1.471G 

0.00234 

"     41 

"     42 

Series  VI. 

Exp.  43 

43  and  45 

1.4827 

0.00473 

1.4785 

0.00255 

1.4744 

0.00046 

"     44 

44,49,    "    48 

1.4706 

0.00737 

1.4694 

0.00655 

1.4681 

0.00584 

«     45 

"     46 

46    "    45 

1.4821 

0.00413 

1.4798 

0.00385 

1.4774 

0.00346 

"     47 

47    "    46 

1.4890 

0.01103 

1.4870 

0.01105 

1.4850 

0.01106 

"     48 

48    "    47 

1.4879 

0.00993 

1.4834 

0.00745 

1.4788 

0.00486 

"     49 

44,49,    "    48 

1.4706 

0.00737 

1.4694 

0.00655 

1.4681 

0.00584 

«     50 

«     51 

"     52 

"     53 

Series  VII. 

Exp.  54 

54  and  55,  58,  63 

1.4715 

0.00647 

1.4696 

0.00635 

1.4677 

0.00624 

"     55 

"     56 

56  and  54 

1.4699 

0.00807 

1.4686 

0.00735 

1.4674 

0.00654 

"     57 

57  and  55,  58,  63 

1.4881 

0.01013 

1.4843 

0.00835 

1.4806 

0.00666 

«     58 

«     59 

59  and  54 

1.4850 

0.00703 

1.4814 

0.00545 

1.4778 

0.00386 

«      60 

60  and  55,  58,  63 

1.4695 

0.00847 

1.4689 

0.00705 

1.4683 

0.00564 

"      61 

61    "    55,58,63 

1.4619 

0.01607 

1.4620 

0.01395 

1.4620 

0.01194 

«      62 

62    "    55,58,63 

1.4711 

0.00687 

1.4691 

0.00685 

1.4671 

0.00684 

"      63 

"      64 

SeriesVIII. 

Exp.  65 

65  and  71 

1.4755 

0.00247 

1.4747 

0.00125 

1.4739 

0.00004 

"      66 

«      67 

"      68 

«      69 

«     70 

70    "    71 

1.4845 

0.00653 

1.4829 

0.00695 

1.4813 

0.00736 

«     71 

Series  IX. 

Exp.  72 

"     73 

"     74 

„ 

Sums  of  the  differences  and 

0.26767 

0.25085 

0.23672 

mean  values  of  a. 

1.47797 

1.47595 

1.47394 

EXPERIMENTS   ON  THE   FLOW  OF   WATER   OVER   WEIRS. 


TABLE     X  —  CONTINUED. 
EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS,  MADE  AT   THE  TREMONT  TURBINE 


33 

34 

35 

Number  of  the 

6  =  0.055 

6  =  0.05. 

6  =  0.045. 

series  and 

of  the 

Differences  from  the 

Differences  from  the 

Differences  from  the 

experiment. 

Values 

mean  value  of  «,  or  from 

Values 

mean  value  of  a,  or  from 

Values 

mean  value  of  a,  or  from 

1.47194. 

1.46994. 

1.46795. 

ota. 

of  a. 

of  a. 

+ 

- 

+ 

— 

+ 

- 

Series  V. 

Exp.   34 

1.4543 

0.01764 

1.4510 

0.01894 

1.4476 

0.02035 

"     35 

1.4706 

0.00134 

1.4695 

0.00044 

1.4684 

0.00045 

"      36 

1.4622 

0.00974 

1.4602 

0.00974 

1.4584 

0.00955 

"      37 

1.4589 

0.01304 

1.4567 

0.01324 

1.4547 

0.01325 

«      38 

"     39 

1.4610 

0.01094 

1.4581 

0.01184 

1.4551 

0.01285 

«      40 

1.4706 

0.00134 

1.4695 

0.00044 

1.4684 

0.00045 

"      41 

«      42 

Series  VI. 

Exp.   43 

1.4703 

0.00164 

1.4662 

0.00374 

1.4622 

0.00575 

"     44 

1.4668 

0.00514 

1.4656 

0.00434 

1.4643 

0.00365 

"     45 

• 

"     46 

1.4751 

0.00316 

, 

1.4728 

0.00286 

1.4705 

0.00255 

a      47 

1.4830 

0.01106 

1.4811 

0.01116 

1.4790 

0.01105 

«     48 

1.4744 

0.00246 

1.4699 

0.00004 

1.4654 

0.00255 

«     49 

1.4668 

0.00514 

1.4656 

0.00434 

1.4643 

0.00365 

"     50 

«     51 

< 

«     52 

"     53 

Series  VII. 

Exp.   54 

1.4657 

0.00624 

1.4638 

0.00614 

1.4619 

0.00605 

"      55 

«      56 

1.4661 

0.00584 

1.4649 

0.00504 

1.4636 

0.00435 

«      57 

1.4769 

0.00496 

1.4733 

0.00336 

1.4696 

0.00165 

«     58 

"      59 

1.4742 

0.00226 

1.4706 

0.00066 

1.4670 

0.00095 

"      GO 

1.4677 

0.00424 

1.4672 

0.00274 

1.4666 

0.00135 

"      61 

1.4620 

0.00994 

1.4621 

0.00784 

1.4621 

0.00585 

"      62 

1.4652 

0.00674 

1.4633 

0.00664 

1.4613 

0.00665 

"     63 

«      64 

Series  VIII. 

Exp.    65 

1.4731 

0.00116 

1.4722 

0.00226 

1.4714 

0.00345 

«      66 

«      67 

"     68 

"     69 

* 

"     70 

1.4797 

0.00776 

1.4782 

0.00826 

1.4765 

0.00855 

«     71 

Series  IX. 

Exp.   72 

"     73 

"     74 

0.23124 

0.22900 

0.23945 

1.47194 

1.46994 

1.46795  i 

96  EXPERIMENTS   ON  THE   FLOW  OF   WATER  OVER    WEIRS. 


EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS,  MADE  AT  THE  CENTRE- VENT  WHEEL  FOR 
MOVING  THE  GUARD  GATES  OF  THE  NORTHERN  CANAL. 

136.  This  centre-vent  wheel  usually  operates  under  about  ten  feet  fall,  and 
is  of  about  sixty  horse-power  under  this  fall.  It  was  constructed  from  nearly 
the  same  designs  as  the  model  centre-vent  wheel,  described  in  art.  100,  and  rep- 
resented on  plate  VII.  For  a  general  description  of  the  Guard  Gates,  see  vol.  I., 
page  775,  Appletoris  Dictionary  of  Machines,  Mechanics,  etc.,  New  York :  D.  Appleton 
&  Co.,  1852. 

A  set  of  experiments  upon  the  power  of  this  wheel  was  made  in  1848,  in 
which  the  water  discharged  by  the  wheel  was  gauged  at  a  weir  constructed  for 
the  purpose,  below  the  wheel.  The  following  experiments  were  made  with  the 
same  apparatus. 

The  total  length  of  the  weir  was  18.02  feet,  which,  for  the  purposes  of 
these  experiments,  was  diminished  to  16.02  feet  by  two  movable  planks  or  par- 
titions, one  foot  wide  each,  the  upstream  faces  of  which,  when  placed  upon  the 
weir,  were  in  the  same  plane  as  the  upstream  face  of  the  weir.  The  form  of 
the  weir  was  such  as  to  give  complete  contraction;  it  was  constructed  of  wood, 
with  the  upstream  face  vertical.  The  crest  of  the  weir  was  formed  of  southern 
hard  pine  plank,  four  inches  in  thickness ;  the  top  was  0.53  inches  wide,  and 
bevelled  off  on  the  downstream  side,  at  an  angle  of  40°  with  the  vertical ;  the 
ends  of  the  weir  and  the  sides  of  the  partitions  were  of  the  same  form. 

The  bottom  of  the  canal  or  basin,  measured  near  the  weir,  was  about  6.72 
feet  below  the  top  of  the  weir.  The  water  discharged  by  the  wheel  passed  to 
the  basin  through  an  irregular  and  contracted  channel,  cut  in  rock,  and  confined 
by  cement  masonry.  This  basin  was  specially  excavated  in  the  rock,  of  large 
dimensions,  in  order  that  the  water  might  reach  the  weir  in  a  sufficiently  quiet 
state  to  permit  a  satisfactory  measurement  to  be  made ;  and  also,  for  the  same 
object,  two  gratings  were  placed  across  the  basin,  parallel  to  each  other,  and 
about  six  feet  apart,  the  downstream  grating  being  about  seventeen  feet  from 
the  weir.  The  effect  of  these  several  precautions  was  such  that,  although  the 
water  escaped  from  the  wheelpit  in  a  rapid  and  turbulent  current,  in  the  basin 
between  the  downstream  grating  and  the  weir,  the  water  was  tranquil  and  free 
from  perceptible  irregularities  in  its  motion  towards  the  weir. 

The  depths  upon  the  weir  were  measured  by  the  hook  gauge,  described  at 
art.  45,  and  represented  by  figures  9,  10,  and  11,  plate  IV.;  this  was  placed  in 


EXPERIMENTS   ON  THE   FLOW   OF   WATER  OVER   WEIRS.  97 

the  basin  about  eight  feet  from  the  weir,  in  a  box,  in  which  the  communication 
with  the  surrounding  water  was  maintained  by  a  small  aperture  in  the  bottom ; 
the  box  and  hook  gauge  were  firmly  attached  to  a  timber  strongly  bolted  to 
the  masonry  forming  one  side  of  the  basin. 

The  quantity  of  water  discharged  by  the  wheel  is  usually  regulated  by  the 
head  gate,  admitting  the  water  from  the  river  into  the  forebay  above  the  wheel. 
When  it  is  desired  to  diminish  the  quantity  discharged  by  the  wheel,  this  gate 
is  partially  closed,  the  effect  of  which  is  to  diminish  the  fall  acting  upon  the 
wheel ;  but  this  method  was  unsuitable  for  these  experiments,  on  account  of  the 
great  agitation  in  the  forebay,  produced  by  the  fall  at  the  head  gate.  During 
these  experiments,  the  head  gate  was  fully  opened,  and  the  quantity  of  water 
discharged  by  the  wheel  was  diminished  by  closing  up  a  portion  of  the  spaces 
between  the  guides,  with  pieces  of  wood. 

The  wheel  was  prevented  from  revolving  by  the  brake  of  the  Prony  dyna- 
mometer. The  entire  apparatus  about  the  wheel  remained  unchanged  throughout 
the  four  experiments,  except  that  the  head  gate  was  closed  on  several  occasions, 
to  enable  the  partitions  on  the  weir  to  be  moved.  This  gate  was  large  (five 
feet  square,)  and  care  was  taken  to  keep  it  open  to  its  full  extent,  in  all  these 
experiments. 

The  apertures  through  which  the  water  entered  the  wheelpit  being  the  same, 
the  quantity  of  water  discharged  must  have  been  uniform,  if  the  head  acting 
upon  the  orifices  had  been  constant;  small  variations,  however,  unavoidably  occurred 
in  the  head,  for  which  it  was  necessary  to  correct  the  depths  upon  the  weir. 
This  has  been  done  in  a  manner  precisely  similar  to  that  adopted  in  .the  experi- 
ments upon  the  weir  at  the  Tremont  Turbine,  described  at  art.  133. 

The  apertures  in  the  wheel  and  between  the  guides,  were  entirely  submerged. 
The  effective  height  of  the  water  in  the  wheelpit  was  measured  in  a  chamber 
constructed  for  the  purpose,  in  the  masonry.  A  free  communication  was  main- 
tained between  the  water  in  the  wheelpit  and  in  the  chamber  by  an  iron  pipe 
about  3.5  inches  diameter.  The  surface  of  the  water  in  the  chamber  was,  in  all 
the  experiments,  above  the  level  of  the  top  of  the  apertures  between  the  guides. 
The  height  above  the  wheel  was  taken  in  the  forebay  nearly  over  the  wheel, 
the  gauge  being  placed  in  a  box  in  the  usual  manner ;  the  zeros  of  the  gauges, 
at  which  both  these  heights  were  taken,  were  at  the  same  level,  consequently, 
the  difference  in  the  readings  gave  the  fall  acting  upon  the  apertures. 


13 


98 


EXPERIMENTS   ON   THE   FLOW   OF   WATER  OVER  WEIRS. 


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EXPERIMENTS   ON   THE    FLOW   OF    WATER  OVER   WEIRS.  99 


EXPERIMENTS  ON  THE  EFFECT  PRODUCED  ON  THE  FLOW  OF  WATER  OVER  WEIRS,  BY   THE 
HEIGHT  OF  THE  WATER  ON  THE  DOWNSTREAM  SIDE. 

137.  These  were  made  at  the  weir  at  the  centre-vent  wheel  for  moving 
the  guard  gates,  with  the  apparatus  used  in  the  preceding  experiments. 

A  singular  phenomenon  was  here  produced,  namely:  under  particular  circum- 
stances, the  flow  of  water  over  a  weir  may  be  increased  by  raising  the  height  of  the  water 
on  the  downstream  side  of  the  weir.  Ordinarily,  when  water  flows  over  a  weir  hav- 
ing contraction  on  the  bottom,  the  under  side  of  the  sheet  near  the  weir,  is 
elevated  above  the  level  of  the  top  of  the  weir,  taking  a  curved  form;  repre- 
sentations of  this  curve  are  given  in  several  works,  the  most  perfect  of  which 
are  by  M.  M.  Poncelet  and  Lesbros,*  who  ascertained  with  great  care  the  forms 
for  several  depths  upon  the  weir.  In  such  cases,  the  space  between  the  sheet 
of  water  and  the  plank  or  other  material  of  which  the  weir  is  composed,  is 
filled  with  air  which  communicates  more  or  less  freely  with  the  external  atmos- 
phere. 

Suppose  the  sheet,  after  passing  the  weir,  to  fall  into  a  body  of  water  of 
considerable  depth,  in  which  the  natural  level  of  the  surface  is  not  very  much 
below  the  top  of  the  weir,  but  sufficiently  so,  as  not  sensibly  to  affect  the  dis- 
charge. The  weir  having  complete  contraction,  the  air  will  remain  under  the 
sheet,  even  if  the  weir  is  of  very  considerable  length  in  proportion  to  the  depth 
flowing  over.  Suppose  now,  that  the  communication  of  the  air  under  the  sheet, 
with  the  external  atmosphere,  is  entirely  cut  off  by  placing  boards  on  the  down- 
stream side  of  the  weir  in  contact  with  each  side  of  the  sheet,  or  by  other 
means,  the  effect  will  ordinarily  be,  that  the  air  under  the  sheet  will  be  wholly 
or  partially  driven  out  by  the  lateral  communication  of  motion  in  fluids,  and  a 
partial  vacuum  will  be  produced,  unless  water  takes  the  place  of  the  air  that  is 
driven  out.  In  either  case,  the  equilibrium  of  the  atmospheric  pressure  on  the 
upper  and  lower  sides  of  the  sheet,  will  be  destroyed,  the  pressure  on  the  upper 
side  preponderating,  the  effect  will  be  to  alter  the  form  of  the  sheet,  and  to 
increase  the  discharge,  by  the  operation  of  forces  bearing  some  resemblance  to 
the  action  in  the  well-known  experiment  with  Venturi's  tube. 

In   the   following    experiments,   this    effect   was   produced    by   raising   the   level 


Experiences  Hydravliques  sur  les  lois  de  I'ecoulement,  etc.   Paris :  1832.  Plate  6. 


100  EXPERIMENTS   ON   THE   FLOW  OF   WATER  OVER  WEIRS. 

of  the  water  on  the  downstream  side  of  the  weir,  to  a  height  a  little  above  the 
top  of  the  weir,  in  consequence  of  which,  by  the  lateral  communication  of 
motion,  the  air  was  driven  out,  and  the  flow  over  the  weir  facilitated. 

During  the  following  experiments,  the  apparatus  was  arranged  in  the  same 
manner  as  in  the  preceding  experiments,  with  these  exceptions,  namely :  the  par- 
titions were  not  used ;  the  quantity  of  water  entering  the  wheelpit  was  diminished 
by  closing  up  more  of  the  spaces  between  the  guides;  the  wheel  was  entirely 
removed ;  and  means  were  provided  for  varying  the  height  of  the  water  on.  the 
downstream  side  of  the  weir.  , 

The  depths  on  the  weir  are  reduced  in  the  same  manner  to  what  they 
would  have  been,  if  the  quantity  of  water  entering  the  wheelpit,  and  flowing 
over  the  weir,  had  been  uniform.  The  details  of  the  experiments  are  given  in 
table  XII. 

That  the  quantity  of  water  entering  the  wheelpit  changed  only  in  a  very 
small  degree  from  any  change  in  the  apparatus,  is  proved  by  the  depths  upon 
the  weir  in  experiments  1  and  9.  The  circumstances  in  both  being  the  same, 
the  corrected  depth  on  the  weir  in  experiment  9,  is  0.0006  feet  less  than  in 
experiment  1,  corresponding  to  a  change  in  quantity  of  about  -g|7  part.  A 
mean  of  the  depths  on  the  weir  in  these  two  experiments  has  been  taken,  with 
which  to  compare  the  other  experiments. 

Measurements  were  also  taken  of  the  thickness  of  the  sheet,  in  the  plane 
of  the  upstream  face  of  the  weir.  This  was  done  by  means  of  a  graduated 
rod  terminating  in  a  fine  point,  and  so  arranged  as  to  slide  in  a  vertical  groove, 
supported  from  one  end  of  the  weir.  These  measurements  were  not  taken  with 
the  same  precision  as  were  the  depths  on  the  weir  with  the  hook  gauge,  prin- 
cipally in  consequence  of  the  oscillations  of  the  surface. 

In  consequence  of  the  want  of  symmetry  in  the  channel  carrying  off  the 
water  from  the  weir,  the  water  on  the  downstream  side  did  not  assume  the 
same  height  at  both  ends  of  the  weir.  Gauges  were  placed  at  both  ends,  pro- 
tected in  a  considerable  degree  from  the  agitation  of  the  water  immediately 
below  the  weir,  and  placed  so  as  to  indicate  the  height  of  the  water  a  short 
distance  downstream  from  the  sheet,  but  the  heights  taken  at  these  gauges  have 
not  the  exactness  of  those  taken  with  the  hook  gauge.  There  were  also  much 
greater  variations  in  the  height  of  the  water  during  the  course  of  an  experi- 
ment, than  occurred  on  the  upstream  side  of  the  weir ;  some  of  the  heights 
given  in  column  9  may  consequently  be  erroneous  to  the  extent  of  0.02  feet. 

The  differences  given  in  column  10,  indicate  the  effect  produced  on  the  dis- 
charge by  the  height  of  the  water  on  the  downstream  side.  When  this  height 


EXPERIMENTS   ON  THE    FLOW   OF   WATEK  OVER    WEIRS.  1Q1 

was  about  3  inches  below  the  top  of  the  weir,  the  effect  was  insensible.  When 
about  level  with  the  top  of  the  weir,  the  obstruction  was  very  minute  and 
barely  sensible.  When  the  height  on  the  downstream  side  was  about  £  of  an 
inch  above  the  top  of  the  weir,  (at  which  height  the  air  did  not  remain  under 
the  sheet,)  the  increase  in  the  discharge  is  quite  sensible,  the  discharge  with  the 
same  depth  being  increased  about  y^.  When  the  height  on  the  downstream 
side  is  1.25  inches  above  the  top  of  the  weir,  the  obstruction  is  quite  distinct, 
and  it  increases  rapidly  with  the  increase  of  height. 


102 


EXPERIMENTS   ON   THE   FLOW   OF   WATER  OVER  WEIRS. 


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EXPERIMENTS   ON   THE   FLOW   OF   WATER  OVER    WEIRS.  103 


EXPERIMENTS   ON  THE   FLOW  OF   WATER   OVER   WEIRS,   MADE  AT   THE  LOWER 

LOCKS,  IN  LOWELL. 

138.  In   the    year    1852,   the    author,   in   connection   with   James    F.   Baldwin, 
Esq.,    the   eminent   engineer    of    Boston,   Massachusetts,   was   employed    to   ascertain 
the    amount    of    water-power    used    by    the    several    manufacturing    companies    at 
Lowell.      In   order   to   be    able   to    do   this   in    a    satisfactory   manner,  it   was   found 
necessary  to    determine   anew  the    rules  for  computing   the  discharge  of  water  over 
weirs   of    certain   forms ;    and   for   this   purpose    an   extensive   series   of    experiments 
was  made,  with   a  very  complete   apparatus,  and  on  a  scale  of  unusual   magnitude. 
The   execution   of    these    experiments   was   intrusted   to    the    author;    and    The   Pro- 
prietors   of   the  Locks    and   Canals   on    McrnmacJc   River,   at   whose    expense   they   were 
made,    have,   with    great    liberality,    given    the    author    permission    to    publish    an 
account   of  them. 

139.  The   great   difficulty   in   this   kind   of  experiment,  usually,  is   to    obtain  a 
suitable   basin   in   which   the   water   flowing   over   the   weir   for   a   certain   period   of 
time  may   be   actually   measured.      Fortunately   for   our  purpose,   the   Lowe*   Locks 
at    Lowell    are    seldom    used,    except    during  the   high   water  in   the   spring,  when 
rafts    can    pass    over    the    rapids    in    the    river    below.      These   locks  were   rebuilt 
principally  of  wood,  in   1841,  and   at  the   time  when   the   experiments  were   made, 
they   were    still    in    good    condition;     they   however    required    some    alterations    to 
adapt   them   to   the    requirements    of    the    experiments;    which   alterations,   together 
with   the    entire  apparatus  employed,  and  the  mode  of  conducting  the  experiments, 
will  now  be  described. 

140.  Plate    XI.,   figure   1,   is  a   general    plan   of    the    Lower    Locks    and    the 
vicinity,   on   a   scale   of  eighty   feet    to    an    inch.      A   is    the    lower  level  of   Paw- 
tucket   Canal ;     B,   the    Eastern    Canal ;     C,    the    Concord    River,   which    enters   the 
Merrimack  Kiver   at   about   1200  feet   below  the   foot  of  the   lock;    D  is  the   dam 
for   discharging   the    surplus   water   from   the   Pawtucket    Canal   into   Concord   River, 
passing   through  the   wasteway   E ;    F,   the  Middlesex  Mills,   which   are    carried    by 
water-power   from   the    Pawtucket   Canal,   through   the    covered    penstock   H;     1,   an 
apparatus    erected  for  the   purpose    of   gauging    the   water   drawn    by   the    Middle- 
sex Mills,  which  was  removed   before  these  experiments  were  made ;    K,  the  upper 
chamber    of    the    lock,  which    was    converted    into    the    gauging    basin    for    these 
experiments,  and  which  is  represented  as  it  was  before  the  alterations  were  made. 


104  EXPERIMENTS  ON  THE   FLOW  OF   WATER  OVER  WEIRS. 

141.  Plate  XII.  represents  the  gauging  chamber  subsequent  to  the  altera- 
tions, on  a  scale  of  10  feet  to  an  inch.  Figure  1  is  a  plan ;  figure  2,  a  lon- 
gitudinal section;  and  figures  3  and  4  transverse  sections.  The  side  wall  A  was 
built  in  1822,  of  large  and  small  stones  laid  without  mortar ;  in  order  to  render 
the  lock  capable  of  holding  water,  it  is  lined  with  planks  about  three  inches 
thick,  secured  by  tree-nails  and  spikes  to  wooden  frames,  which  are  supported  on 
the  bottom  by  the  earth  and  some  rough  walls,  and  on  the  sides  by  the  side 
walls  of  the  lock.  As  originally  constructed,  the  planking  was  fastened  to  posts 
resting  immediately  against  the  side  walls ;  but  when  reconstructed  in  1841,  the 
chambers,  together  with  the  gates,  were  narrowed  to  the  width  represented,  which 
is  about  one  half  the  former  width;  at  that  time,  also,  the  parts  B B,  about 
the  hollow  quoins,  were  built  anew  in  cut  granite,  laid  in  hydraulic  cement. 

To  prepare  the  chamber  for  these  experiments,  the  upper  set  of  lock  gates 
and  the  corresponding  mitre  sills  were  removed,  and  the  weir  C,  plate  XII., 
figures  1  and  2,  constructed  in  place  of  them ;  the  middle  gates  were  also 
removed,  and  the  lower  end  of  the  chamber  closed  with  timbers  and  plank, 
as  represented  at  D;  in  the  lower  part  of  this  timber  work  the  waste  gate  K 
was  constructed,  for  the  purpose  of  drawing  off  the  water  from  the  chamber, 
after  each  experiment. 

The  construction  of  the  wooden  sides  of  the  chamber  was  such,  that  when 
the  chamber  was  partially  or  wholly  filled  with  water,  they  would  yield  a  little 
to  the  pressure,  and  the  capacity  would,  consequently,  be  increased  beyond  what 
it  was  when  empty,  which  was  necessarily  the  case  when  the  dimensions  were 
taken.  To  diminish,  as  much  as  practicable,  this  source  of  error,  the  braces  E 
were  placed  across  the  chamber,  just  above  the  water-line  F '  F,  nearly  up  to 
which  the  chamber  was  filled  in  the  experiments.  These  braces  were  placed 
opposite  each  side  timber  in  the  frame  of  the  chamber,  excepting  at  G  Gf,  where 
a  flooring  of  thick  plank,  put  in  for  another  object,  answered  the  same  purpose ; 
afterwards,  every  accessible  timber  in  the  sides  was  strongly  braced  and  keyed  up 
from  the  side  walls,  which  was  done  with  such  force,  that  the  ends  of  the  braces 
E  were  indented  into  the  planks  forming  the  sides  of  the  chamber.  At  Hit, 
where  the  space  between  the  walls  and  the  planking  was  too  small  to  admit  of 
the  bracing,  the  spaces  between  the  timbers  were  filled  with  small  stones,  dropped 
and  rammed  in  from  the  top.  These  operations  stiffened  the  sides  of  the 
chamber  so  much,  that  the  correction  required  for  the  enlargement  of  the  capac- 
ity of  the  chamber,  in  consequence  of  the  yielding  of  the  sides,  was  very  minute. 
All  the  leakages  that  could  be  detected  were  stopped  by  various  contrivances ; 
the  depressions  in  the  planks,  about  the  heads  of  the  spikes,  were  filled  up  with 


EXPERIMENTS   ON   THE   FLOW  OF   WATER  OVER  WEIRS.  1Q5 

cement ;  the  sides  of  the  planking  towards  the  chamber  were  scraped,  and  ren- 
dered as  smooth  and  uniform  as  practicable. 

A  part  of  the  wall  A  was  removed  at  I,  for  the  purpose  of  discharging  the 
water,  flowing  over  the  weir,  directly  into  the  wasteway,  whenever  it  was  neces- 
sary to  divert  its  flow  from  the  chamber;  the  floor  GG  was  continued  through 
the  wall,  as  represented  in  figures  1  and  4,  plate  XII. 

142.  Plate  XIII.,  figure  1,  is  a  longitudinal  sectional  elevation  through  the 
middle  of  the  weir,  showing  most  of  the  apparatus  immediately  connected  with 
it.  A  is  a  plate  of  cast-iron  forming  the  crest  of  the  weir ;  it  is  ten  feet  long, 
thirteen  inches  wide,  and  an  inch  thick,  accurately  and  smoothly  planed  in  every 
part ;  the  upper  corner  presented  to  the  current  is  square  and  sharp,  or  as 
nearly  so  as  cast-iron  can  be  conveniently  maintained ;  the  horizontal  part  of 
the  top  is  0.25  inches  wide ;  the  remainder  of  the  top  is  bevelled  off  at  an 
angle  of  45° ;  this  plate  is  secured  to  the  timber  work  by  numerous  screws 
with  countersunk  heads;  the  timber  work  is  strongly  bolted  to  the  granite  hol- 
low quoins  of  the  lock.*  The  ends  of  the  weir  B  are  formed  of  plates  of  cast- 
iron,  of  similar  section  to  the  plate  A.  The  whole  upstream  side  of  the  weir 
forms  a  vertical  plane  13.96  feet  in  length,  and  4.60  feet  in  depth,  from  the  top 
of  the  plate  A  to  the  top  of  the  masonry  C;  the  upstream  side  of  the  plates 
B  are  also  in  the  same  vertical  plane. 

D  is  the  swing  gate  for  admitting  and  diverting,  at  will,  the  stream  of 
water  flowing  over  the  weir,  into  or  from  the  measuring  chamber  E.  FFF 
are  leak  boxes  or  troughs,  to  catch  the  leakage  by  the  edges  of  the  swing 
gate,  when  shut.  The  water  thus  caught  is  conveyed  to  openings  G,  cut  through 
the  planking  on  each  side  of  the  chamber,  through  which  it  is  discharged,  thus 
preventing  any  embarrassment  from  the  leakage  of  the  swing  gate  when  shut, 
as  it  does  not  enter  the  chamber  E.  The  swing  gate  is  suspended  from  the 
pivots  H;  all  its  parts  are  made  as  light  as  practicable,  consistent  with  the 
required  stiffness,  in  order  that  the  time  occupied  in  opening  or  shutting  it  may 
be  as  short  as  possible.  A  very  important  part  of  the  experiments  consisted  in 
determining  the  length  of  time  during  which  the  water  flowed  into  the  meas- 
uring chamber  E;  this  was  obtained  by  observing  the  time  when  the  swing 
gate  was  opened  and  shut,  which  was  done  by  an  observer  in  the  building  f, 
by  means  of  an  electric  telegraph  and  a  marine  chronometer,  in  the  following 
manner.  The  break  circuit  apparatus  K  is  fixed  in  such  a  position  that,  when 
one  half  only  of  the  stream  flowing  over  the  weir  passes  into  the  chamber,  the 
cam  L,  attached  to  the  frame  of  the  swing  gate,  depresses  the  knob  as  repre- 

14 


106  EXPERIMENTS   ON  THE   FLOW   OF   WATER  OVER   WEIRS. 

sented  in  the  plate,  and  breaks  the  circuit  of  the  electric  current  in  the  wire 
M;  this  causes  a  sound  to  be  made  by  the  call  N,  in  the  small  building  7, 
where  sits  the  observer  with  his  eye  on  the  chronometer,  who  notes  the  time 
when  the  sound  is  made ;  the  chronometer  used  beats  half  seconds,  but,  by 
employing  a  practised  observer,  the  time  was  noted  to  tenths  of  a  second,  the 
error  probably  rarely  exceeding  two  tenths  of  a  second.  The  gate,  with  its 
accompanying  apparatus,  was  balanced,  so  that  it  could  be  opened  or  shut  with 
sensibly  the  same  amount  of  force ;  this  balancing  was  done  with  the  water  flow- 
ing over  the  weir,  and  was  done  anew  for  each  material  variation  in  the  quan- 
tity. To  each  of  the  timbers  0  and  P,  plate  XIII.,  figure  1,  was  attached,  by 
a  joint,  a  prop  L,  shown  at  figure  4,  plate  XII.;  the  prop  at  the  timber  0, 
for  the  purpose  of  retaining  the  swing  gate  in  its  position  when  open,  and  the 
other  at  the  timber  L,  to  retain  it  in  position  when  shut.  The  movement  of 
the  gate  was  produced  by  placing  weights  upon  the  frame  at  Q  and  R,  plate 
Xin.,  where  the  gate  is  represented  as  at  the  middle  point  of  its  motion  while 
shutting;  the  motion  being  produced  by  the  gravitation  of  the  weights  at  Q. 
As  soon  as  the  gate  is  shut,  the  prop  L,  plate  XII.,  figure  4,  is  placed  under 
the  frame  at  R,  plate  XIII.,  and  keyed  up  tight;  the  weights  are  then  taken 
off  at  Q)  and  about  the  same  amount  of  weight  is  placed  at  R;  then,  when  it 
is  desired  to  open  the  gate,  an  assistant  strikes  the  prop  from  under  R  with  a 
sledge-hammer,  when  the  weight  at  R  causes  the  gate  to  open ;  the  prop  is 
then  immediately  placed  under  the  frame  at  Q.  To  prevent  injurious  concussions 
from  the  action  of  the  weights,  thick  pieces  of  India-rubber,  operating  as  springs, 
were  fastened  on  the  under-side  of  the  frame  at  Q  and  R,  which,  when  the  gate 
attained  either  of  its  extreme  positions,  struck  upon  the  corresponding  stops  $ 
and  T. 

From  the  foregoing  description  of  the  apparatus,  the  manner  of  operating  the 
swing  gate  will  be  readily  understood.  Four  assistants  were  employed  for  the 
purpose.  Suppose  that  -the  chamber  is  nearly  filled,  and  that  it  is  required  that, 
when  the  water  reaches  a  certain  height,  the  flow  of  the  water  shall  be  diverted 
from  the  chamber:  one  assistant,  who  has  been  watching  the  rise  of  the  water, 
gives  a  signal  when  the  water  has  reached  the  desired  height,  at  which  the  prop 
under  the  frame  at  Q  is  immediately  knocked  away,  the  weights  at  Q  cause  the 
gate  to  move  until  it  strikes  the  weir,  or  the  India-rubber  springs  strike  the  stops 
T;  at  that  moment  another  assistant  places  the  prop  under  R,  and  the  flow  of  the 
water  is  diverted  from  the  chamber;  another  assistant  then  changes  the  weights, 
and  the  apparatus  is  ready  for  the  reverse  operation  by  which  the  gate  is 


EXPERIMENTS   ON   THE   FLOW   OF   WATER  OVER   WEIRS.  107 

opened.  Much  time  was  occupied  in  adjusting  this  apparatus  so  that  the  cam  Z, 
plate  XIII.,  would  strike  the  break  circuit  when  the  gate  was  in  such  a  position 
that  one  half  of  the  water  flowing  over  the  weir  passed  into  the  chamber ;  and 
also,  that  the  time  in  which  the  gate  moved  through  each  half  of  the  thickness 
of  the  sheet,  would  be  the  same.  It  required  a  new  adjustment  for  each  depth 
upon  the  weir.  Precise  accuracy  was  not  attained  or  attempted,  in  any  of  these 
adjustments,  but  such  an  approximation  was  made,  that  it  is  believed  that  the 
errors  arising  from  want  of  complete  exactness,  are  entirely  insensible  in  the  results. 
143.  The  depths  upon  the  weir  were  observed  by  means  of  the  hook  gauges 
U  and  V,  plate  XII.,  figures  1  and  2,  and  plate  XIII,  figure  1.  One  of  these 
gauges  is  represented  in  detail  by  figures  2,  3,  and  4,  plate  XIII.,  £  the  full  size. 
They  were  made  by  the  Lowell  Machine  Shop.  This  valuable  instrument  has 
been  sufficiently  described  in  the  account  of  the  experiments  on  the  Tremont 
Turbine  (art  45).  These  gauges  were  placed  in  wooden  boxes  closed  on  all  sides, 
excepting  at  the  top ;  in  the  bottom  of  each  of  which  was  a  hole  about  an 
inch  in  diameter,  and  in  that  part  of  the  bottom  projecting  beyond  the  lines  of 
the  canal  walls,  due  care  being  taken  that  the  plugs,  by  which  the  holes  were 
•  partially  closed,  did  not  project  through  the  bottom.  In  the  experiments  on  the 
weir  in  which  the  end  contraction  was  suppressed,  a  communication  was  established 
between  the  gauge  boxes  and  the  canal  leading  to  the  weir,  by  pipes  opening  at 
B,  figures  8,  9,  and  10,  plate  XIV.  The  pipes  opening  near  the  bottom  of  the 
canal,  six  feet  from  the  weir,  forming  part  of  the  system  for  taking  the  heights 
at  different  distances  from  the  weir,  were  also  used  in  some  of  the  experiments. 
The  boxes  were  securely  fastened  to  wooden  posts  in  the  angles  of  the  gate 
recesses ;  and  the  posts  were  strongly  fastened  to  the  walls,  by  several  iron  bolts 
driven  into  holes  drilled  in  the  granite  stones  for  the  purpose.  It  was  very 
important  that  these  gauges  should  be  immovably  fixed,  relatively  to  the  weir. 
It  is  probable,  however,  that  they  were  not  perfectly  firm.  During  the  course  of 
the  experiments,  two  comparisons  were  made  of  the  relative  heights  of  the  gauges 
and  the  top  of  the  weir ;  one  on  October  26th ;  the  other,  November  8th,  when 
there  was  found  to  be  a  sensible  difference  in  them,  the  most  probable  cause 
of  which  was,  that  changes  took  place  in  the  absolute  height  of  the  gauges, 
that  did  not  affect  the  weir  in  the  same  degree.  It  is  difficult  to  perceive  how 
the  weir  could  change  from  a  settlement  of  the  masonry,  founded,  as  it  is,  on 
rock;  the  walls  to  which  the  gauges  were  attached,  were  much  less  substantially 
built  and  not  founded  on  rock ;  it  is  not  impossible  that  changes  took  place  in 
the  timber-work  of  the  weir,  by  the  absorption  of  water,  notwithstanding  it  was 


108  EXPERIMENTS   ON  THE   FLOW  OF   WATER  OVER   WEIRS. 

fixed  in  place  several  weeks  before  the  experiments  were  made,  with  a  view  to 
its  complete  saturation. 

If  the  apparatus  is  sufficiently  stable,  the  comparison  of  the  heights  of  the 
hook  gauges  with  the  top  of  the  weir,  can  be  made  with  any  desired  degree  of 
precision.  For  making  the  comparison  in  these  experiments,  the  following  appara- 
tus was  devised.  The  water  being  drawn  out  of  the  canal,  the  top  of  the  weir 
was  inclosed  in  a  water-tight  trough,  containing  only  a  small  quantity  of  water, 
but  sufficient  to  cover  the  crest  of  the  weir  to  a  small  depth ;  this  trough  was 
connected  with  the  hook  gauge  boxes,  by  leaden  pipes;  the  boxes  were  rendered 
water-tight  by  coating  the  joints  with  pitch,  and  plugging  up  the  holes  in  the 
bottom ;  they  were  also  carefully  propped  up.  The  communication  being  free, 
and  the  leakages  very  small,  the  water  on  the  crest  of  the  weir  and  in  the 
boxes,  would  stand  at  the  same  level ;  consequently,  all  that  remained  to  be 
done,  was  to  measure  the  height  of  the  water  with  the  hook  gauges,  and,  at  the 
same  time,  the  depths  upon  the  crest  of  the  weir.  The  measurement  by  the 
hook  gauges  presented  no  difficulty,  as  it  required  nothing  more  than  the  ordi- 
nary use  of  the  instruments.  To  measure  the  depths  upon  the  crest  of  the 
weir,  had  always  been  a  difficulty  in  making  similar  comparisons ;  to  meet  it  in 
this  case,  the  instrument,  represented  at  plate  XIII.,  figure  5,  was  devised.  The 
points  were  numbered  from  1  to  10,  and  the  exact  height  of  each  of  them 
above  a  horizontal  plane,  on  which  the  instrument  stood,  was  ascertained.  In 
using  this  instrument,  the  water  in  the  trough  was  adjusted  to  a  convenient  level; 
the  top  of  the  weir  was  divided  into  ten  equal  spaces;  the  instrument  was 
placed  upon  one  of  them,  and  when  the  water  became  quite  tranquil,  the  num- 
ber of  the  point  that  coincided  with  the  surface  was  noted,  and,  at  the  same 
moment,  the  heights  of  the  water  in  the  boxes  were  observed  with  the  hook 
gauges.  If  (as  was  usually  the  case)  the  surface  of  the  water  did  not  exactly 
coincide  with  either  of  the  points,  the  true  fractional  number  was  taken  by  esti- 
mation. The  adjacent  points  differed  in  height  about  0.001  feet,  and  a  fourth 
part  of  this  quantity  was  sufficiently  distinct  not  to  be  doubtful. 

As  an  example  of  the  precision  attainable  by  the  use  of  this  instrument,  the 
following  results  are  given  of  the  comparison  of  the  north  hook  gauge  with  the 
weir,  made  during  the  night  of  October  26,  by  Mr.  John  Newell.  The  results 
indicate  the  corrections  to  be  applied  to  the  reading  of  the  hook  gauge,  to  give 
the  true  height  of  the  surface  of  the  water  in  the  gauge  box,  above  the  top 
of  the  weir,  each  result  being  a  mean  of  eleven  measurements  made  at  equidis- 
tant points  on  the  weir. 


EXPERIMENTS   ON  THE   FLOW   OF   WATER  OVER  WEIRS. 


109 


By  the  1st  trial,  the  correction  was — 0.03076  feet. 

-0.03032     « 

—  0.03076    « 

—  0.03096     « 


u 

2d      « 

u 

u 

u 

3rd     " 

u 

cc 

u 

4th     " 

u 

u 

u 

5th     " 

a 

u 

Mean 

—  0.03072  feet. 

The  extreme  variation  is  between  the  2nd  and  4th  trials,  amounting  to  0.00064 
feet,  a  quantity  scarcely  visible  to  the  naked  eye ;  of  course,  in  the  mean  result 
of  all  the  trials,  the  error  of  observation  must  be  entirely  insensible. 

It  has   been   remarked   that   the   comparisons   made    at   different   times,  did   not 
give   the   same   results.     Two   complete   comparisons  were   made,  as  follows :  — 


DATE, 
1852. 

CORRECTIONS. 

North  hook  gauge. 
Feet. 

South  hook  gauge. 
Feet. 

October  26th. 
November  8th. 

—  0.03072 

—  0.03250 

—  0.02786 

—  0.03069 

> 

Considering  the  care  with  which  these  comparisons  were  made,  and  the  per- 
fection of  the  method,  the  differences  cannot  be  attributed  to  errors  of  observa- 
tion, but,  rather,  to  a  want  of  stability  in  some  parts  of  the  apparatus.  The 
corrections  determined  October  26th,  were  used  in  reducing  all  the  experiments 
made  from  October  20th,  to  November  7th,  both  inclusive ;  for  all  subsequent 
experiments,  the  corrections  found  November  8th  were  used. 

The  twenty-three  experiments  numbered  from  11  to  33,  in  table  XIII.,  were 
made  under  circumstances  as  nearly  identical  as  practicable.  They  were  made  at 
different  times  throughout  the  course,  for  the  purpose  of  neutralizing  errors  of 
the  same  class  as  that  just  described,  the  resulting  effects  of  which  ought  to  be 
shown  by  the  variation  in  the  coefficients  deduced  from  experiments  made  at 
different  times.  These  experiments  are  collected  together  in  the  following  table:  — 


Differences  from  the 

DATE, 

Number  of 

mean  deduced  from  all 

experi- 

Mean coefficients. 

the  experiments,  or 

1862. 

ments. 

from  3.3223. 

Oct.  20th,  P.M.,  and  Oct.  21st,  A.M. 

6 

3.3186 

—  0.0037 

Oct.  21st,  p.  M.,  and  Oct.  22d,  A.M. 

8 

3.3216 

—  0.0007 

•Oct.  29th,  P.M. 

6 

3.3278 

+  0.0055 

Nov.  llth,  P.M. 

3 

3.3207 

—  0.0016 

EXPERIMENTS   ON  THE   FLOW   OF   WATER  OVER  WEIRS. 

The  extreme  variation  is  between  the  experiments  of  October  20th  and  29th, 
in  which  it  amounts  to  ¥£T.  The  greatest  difference  from  the  mean  deduced 
from  all  the  23  experiments,  is  in  the  coefficient  deduced  from  the  experiments 
of  October  29th,  in  which  it  amounts  to  F|T.  It  is  fair  to  presume  that  simi- 
lar irregularities,  not  in  any  case  much  exceeding  the  above,  and  arising  princi- 
pally from  want  of  stability  in  the  apparatus,  exist  in  other  parts  of  this  series 
of  experiments. 

144.  The    capacity   of    the   gauging    chamber   was   obtained    by  measuring    its 
dimensions.      For   this   purpose,   horizontal    lines    were    traced   on   the   sides   of    the 
chamber    at    every    foot    in    height ;     the    widths    were    then    measured    at    right 
angles  to  the  sides,  at  points  two  feet  apart;    from  these  widths,  and   other   neces- 
sary measurements,  the   total   area  was   obtained  at   each   horizontal   section.     When 
these    measurements   were    made,    the    chamber   was    of    course    empty,    but   when 
filled    with   water,    its    dimensions   would   evidently   be    somewhat    larger,    in    conse- 
quence   of    the    sides    and    bottom    yielding    to    the    pressure.      To    ascertain   what 
allowance    to    make   for   this,    a    systematic    measurement  was    made   in   the   spaces 
between    the    planking    and    the   walls,   both   when    the    chamber   was    empty   and 
when   filled   to  the    usual   height;   similar   measurements  were  made  for   the  bottom, 
by   placing  poles   vertically,   resting   upon,  and   fastened   to  the   bottom ;    the    eleva- 
tions  of   the    tops   of   these    poles  were   taken   with    a    levelling    instrument,  both 
when  the  chamber  was  empty,  and  when  filled.      It  was  thus  ascertained   that  the 
capacity   of  the   chamber,   when   filled   with   water   to    the    usual   height,   was   11.11 
cubic  feet  greater  than  when  empty. 

Two  persons  made  independent  measurements  of  the  capacity  of  the  chamber, 
the  results  of  which  differed  only  about  £  of  a  cubic  foot,  a  coincidence  which 
must  of  course  be  considered  as  accidental.  The  capacity  finally  determined  upon 
for  9.5  feet  in  height,  (which  was  nearly  the  depth  filled,  in  each  experiment,) 
and  including  the  enlargement  resulting  from  the  pressure,  was  12138.18  cubic 
feet. 

145.  The   chamber  was  not  quite  water-tight,  but  the   amount  of  the   leakage 
was   determined    by   noting  the   rate   at   which   the    surface    of   the   water    lowered, 
when   none   was   admitted    from    the   weir,    and    the    waste    gate    was   closed ;    this 
was   repeated   with   the   water  in    the    chamber    at    different   depths.      It  was   thus 
found    that    the    mean    leakage   was    0.035    cubic    feet    per    second ;     that    is,   the 
product    of    0.035   multiplied    by   the    number    of    seconds    that    the   water   flowing 
over    the   weir    continued   to   enter   the   chamber   during    an    experiment,   must    be 
added   to   the   quantity   in   the    chamber    at    the    moment    the   water   was   diverted, 
in  order  to  give  the  true  quantity  that  passed  over  the  weir  in  the  same  time. 


EXPERIMENTS   ON  THE   FLOW  OF   WATER  OVER   WEIRS.  HI 

146.  It  was  not  convenient  to  empty  the  chamber  entirely  after  each  experi- 
ment,  but  the    heights   of    the   water   in    the   chamber    at    the   beginning   and    end- 
ing   of    each,    were    ascertained   with    great    accuracy   by   means    of    hook    gauges, 
placed  in  the  boxes  X  and   Y,  figures  1,  2,  and  3,  plate  XII.,  which  were  fastened 
to   a   post   strongly   bolted    to    the    wall   A.      A   communication   was    established,   at 
will,   between   the   water   in   the    chamber    and   either   of    the    boxes,   by   pipes   and 
cocks.      The   operation   of  taking   the    heights  was   as   follows :    the  chamber   having 
been    sufficiently   emptied,   the    waste    gate   K  was    closed,   the    communication    of 
the   lower    box    with    the    chamber   was    established,   and   when    the   oscillations   in 

*• 

the  surface  had  ceased,  the  height  of  the  water  was  taken;  the  cock  was  then 
shut,  and  a  signal  made  for  opening  the  swing  gate.  When  the  chamber  had 
been  filled,  and  the  flow  of  water  into  the  chamber  diverted  by  closing  the 
swing  gate,  the  communication  with  the  upper  box  was  opened ;  when  the  oscil- 
lations had  ceased,  observations  of  the  water  were  taken  at  short  and  regular 
intervals,  for  some  minutes,  the  time  and  height  being  noted.  In  consequence  of 
the  leakage  of  the  chamber,  the  surface  lowered  slowly,  and  the  continued  obser- 
vations were  made  for  the  purpose  of  being  able  to  infer  the  exact  height  at 
which  the  water  stood  in  the  chamber  at  the  instant  that  the  swing  gate  was 
shut,  the  very  slow  rate  at  which  the  surface  of  the  water  in  the  chamber 
lowered,  permitting  this  to  be  done  with  great  precision.  For  the  success,  how- 
ever, of  this  operation,  it  was  essential  that  the  timekeeper  used  should  agree 
with  the  chronometer,  by  which  the  times  of  opening  and  shutting  the  swing 
gate  were  noted ;  it  was  accordingly  frequently  compared,  and  any  difference 
noted. 

147.  Plate  XIV.  represents  the   different  forms   of  weir  on  which  experiments 
were   made.     All   the  figures   are   on   the   same   scale,  namely,  five  feet   to   an   inch, 
or  -fa  the  full  size. 

Figure  1  is  a  longitudinal  section,  figure  2,  a  plan,  and  figure  3,  an  eleva- 
tion of  what  we  call  the  regular  weir,  that  is,  a  weir  in  which  the  contraction  is 
complete,  both  on  the  ends  and  on  the  bottom. 

Figure  4  is  an  elevation  of  a  weir  of  precisely  the  same  form  as  that  last 
described,  excepting  that  it  is  divided  into  two  equal  parts  or  bays  by  the  par- 
tition, which  is  two  feet  wide.  The  upstream  side  of  the  partition  is  in  the  same 
vertical  plane  as  the  remainder  of  the  weir,  having  no  bolt  heads  or  other  pro- 
jection below  the  level  of  the  surface  of  the  water. 

Figures  5,  6,  and  7,  represent  a  weir  of  precisely  the  same  form  as  that 
first  above  described,  excepting  that  the  depth  of  the  canal  approaching  the  weir 
is  diminished. 


112  EXPERIMENTS  ON  THE   FLOW   OF   WATER  OVER   WEIRS. 

Figures  8,  9,  and  10,  represent  the  same  weir  as  first  above  described,  modi- 
fied so  that  the  contraction  at  the  ends  is  suppressed,  that  is,  the  canal  leading 
to  the  weir  is  of  the  same  width  as  the  weir.  These  figures  also  show  the 
apparatus  used  to  ascertain  the  effect  of  taking  the  depths  upon  the  weir  at 
different  distances  from  it,  by  means  of  pipes  opening  near  the  bottom  of  the 
canal. 

Figures  11  and  12  represent  the  upper  part  of  a  dam,  of  the  same  section 
as  that  erected  by  the  Essex  Company,  in  1846-8,  across  the  Merrimack  Eiver 
at  Lawrence,  (about  nine  miles  below  Lowell).  This  magnificent  work  has  an 
overfall  900  feet  in  length,  the  perpendicular  fall  being  about  24  feet.  This 
form  was  experimented  upon,  in  order  to  obtain  a  formula  for  computing  the 
flow  of  the  river  over  this  dam. 


DESCRIPTION  OF  TABLE  XIII. 

Containing  the   details   of  the   experiments   on  the  flow  of  water  over  weirs,  made  at  the  Lower  Locks, 

Lowell,  in   October  and  November,  1852. 

148.  The  columns  numbered  from  1  to  5,  require   no  further  explanation  than 
is  contained   in   the   respective    headings. 

149.  COLUMN  6.     Duration  of  the  experiment.     This  is  the  interval  of  time  during 
which  the  water  flowed   into  the  chamber ;    it  is  obtained   by  taking  the  difference 
of  the  corresponding  times  in  column  5. 

150.  COLUMN  7.     Mean  depth  upon  the  weir  ly  observation.     It  was   found   imprac- 
ticable   in  many   cases,   to   maintain   the   canal   at   a   uniform   height   throughout   an 
experiment,   although    every    endeavor  was    made.      For    instance,   no    experiments 
were   made   when    the    mills  were    in    operation,   nor    until    some    hours   after   the 
usual    time   when    they   ceased   drawing   water;    this   rendered   it  necessary   to   per- 
form the   experiments   either  during   the    night,  or   on   Sunday ;    in  consequence   of 
the    lateness    of    the    season,    advantage    was    taken    of    both    these    opportunities. 
When   any   change   was    made    in    the    level    of   the   water    in   the   canal,  for   the 
purpose  of  varying   the  depths  upon  the  weir,  a   considerable   time  was   allowed   to 
elapse  before   the  experiments  were  resumed,  in   order  that   the   level   of  the  water 
might   get  well    established.      In   spite    of    all    precautions,    however,    variations    fre 
quently  occurred  in   the    depths   upon   the  weir,  which,  with   the    ordinary  mode   of 
taking  an  arithmetical  mean   of  the   several  observations   of  the   depth,  would   have 
materially   affected   the   accuracy   of  the   results ;    this   difficulty   was   obviated   in   a 


EXPERIMENTS   ON   THE   FLOW  OF   WATER  OVER  WEIRS.  H3 

great   degree,  by   the   use   of  a  novel   mode   of  obtaining   the   mean   depth,   which 
will  now  be  explained.     Let 

h,h',h",Qic.hn  represent    the    several    observed    depths    upon    the    weir,   the    suc- 

cessive values  not  differing  greatly  from  each  other. 

t,t',t",etc.tn,  the  corresponding  intervals  of  time  between  the  several  observations; 
T,  the  sum  of  all  the  intervals  of  time  ; 
Q,  the   total   volume   of  water  actually   flowing   over   the   weir    in  the 

time  T; 
If,  the   mean   depth   upon   the   weir   that  would   discharge   the   volume 

Q,  in  the  time  T; 

• 

I,  the  length  of  the  weir  ; 
C,  a  constant  coefficient  : 

we  shall  have,  evidently,  very  nearly, 

"*+  etc.  + 


we  have  also 


whence  we  derive,  by  substituting  the  value  of   Q  previously  found, 


As  an  example  of  the  application  of  this  method,  let  us  take  the  observations 
made  at  the  north  hook  gauge  during  experiment  74  ;  this  is  selected,  simply 
because  the  variations  in  the  depths  upon  the  weir  were  greater  than  in  any 
other  experiment. 

15 


114 


EXPERIMENTS   ON  THE   FLOW  OF  WATER  OVER   WEIRS. 


EXTRACT  FROM   THE   NOTES   TAKEN  AT   THE  NORTH   HOOK  GAUGE. 


Commencement  of  the  experi- 
ment by  the  time  of  this  watch. 
9*  12' 12.4". 


Ending  of  the  experiment  by 
the  time  of  this  watch,  9*24' 
49.9". 


October  24th,  1852,  A.*., 

9»,  V,  watch  12"  fast. 

TIME. 

Beading  of  the 
hook  gauge. 

9*  9'  15" 

0.6360 

10    50 

0.6320 

11    45 

0.6325 

12    45 

0.6310 

1 

13    15 

0.6310 

1 

14    20 

0.6300 

1 

14   50 

0.6365 

3 

15    20 

0.6290 

1 

16    30 

0.6300 

1 

17     5 

0.6335 

2 

17    55 

0.6380 

3 

18    35 

0.6480 

5 

19    20 

0.6500 

6 

20     0 

0.6470 

5 

20    55 

0.6470 

5 

21    25 

0.6445 

4 

22    10 

0.6530 

7 

22    35 

0.6550 

7 

23     5 

0.6480 

5 

23    45 

0.6580 

8 

24   35 

0.6605 

9 

Arithmetical    }  0>6428 
mean  reading, ) 

For  the  purpose  of  simplifying  the  operation  of  finding  the  mean,  it  is 
assumed  that  we  can,  without  sensible  error,  use  an  arithmetical  mean  of  all 
depths  not  varying  more  than  0.002  feet  from  each  other ;  accordingly  an  arith- 
metical mean  has  been  taken  of  all  the  readings  marked  1  in  the  margin  of 
the  above  table,  and  similar  means  have  been  taken  of  the  other  readings  marked 
with  the  same  number  in  the  margin.  It  will  be  perceived  that  it  was  noted 
at  9h  5',  that  the  watch  was  12"  fast ;  by  another  comparison  with  the  chronom 
eter  made  at  10h  47',  the  watch  was  22"  fast;  from  these  two  comparisons  it  is 
inferred  that,  at  the  middle  of  the  experiment,  the  watch  was  13.3"  fast.  Instead 
of  changing  the  times  of  all  the  observations,  the  time  of  the  commencement  and 
ending  of  the  experiment  has  been  changed  to  conform  to  this  watch,  but  for 
the  purpose  of  this  reduction  only.  By  the  method  adopted,  it  is  assumed  that 
the  height  of  the  water  did  not  change  until  half  the  interval  of  time  between 
two  consecutive  observations  had  elapsed ;  accordingly,  we  find  that  the  time  cor- 


EXPERIMENTS   ON  THE   FLOW   OF   WATER  OVER  WEIRS. 


115 


responding  to  the  first  mean  depth,  is  from  the  beginning  of  the  experiment  to 
9h  14'  35",  or  142.6",  and  from  9"  15'  5"  to  9h  16'  47.5",  or  102.5",  making  245.1". 
The  several  mean  readings  and  the  corresponding  times,  given  in  the  following 
table,  are  obtained  in  this  manner;  the  depths  upon  the  weir  corresponding  to  the 
several  mean  readings,  are  also  given,  which  are  found  by  subtracting  0.03072  feet 
from  each  mean  reading,  (see  art.  143). 


Time  correspond- 

Mean depths  upon 

Number 

Mean  reading  of  the 

ing  to  each  mean 

the  weir,  deduced 

of  the 

hook  gauge. 

reading. 

from  the  several 

mean 

mean  readings. 

reading. 

Feet. 

Seconds. 

Feet. 

i 

0.63020 

245.1 

0.59948 

2 

0.63350 

42.5 

0.60278 

3 

0.63725 

75.0 

0.60653 

4 

0.64450 

37.5 

0.61378 

5 

0.64750 

167.5 

0.61678 

6 

0.65000 

42.5 

0.61928 

7 

0.65400 

62.5 

0.62328 

8 

0.65800 

45.0 

0.62728 

9 

0.66050 

39.9 

0.62978 

The  quantities  in  the  third  column  of  this  table  are  the  values  of  *-,  -i- ,  etc., 

i         &    ' 

in  the   expression    given  above   for   H;  the   quantities  in    the   fourth    column    are 

the   corresponding    values    of   h,  h',   etc.  The    value    of    T   being    757.5,   all    the 

quantities  in  the  second  member  of  the  equation  are  known;  by  substituting  these 
values  we  find 

H=  0.6113. 

The  arithmetical  mean  of  the  eighteen  observations  is  0.6428 ;  deducting  the 
correction  0.03072,  we  find  the  mean  depth  to  be  0.6121 ;  the  difference  by  the 
methods  is  0.0008. 

A   similar   computation   on   the   observations  at  the   south  hook   gauge   gives 

H=  0.6099. 

By  taking  the  arithmetical  mean  of  the  observations,  we  find  the  depth,  by  the 
south  hook  gauge  to  be  0.6096. 

The  mean  of  the  above  values  of  H,  or  0.6106,  is  adopted  as  the  depth  on 
the  weir  in  experiment  74. 

A  similar  reduction  has  been  made  of  the  observations  at  each  hook  gauge, 
in  all  the  experiments ;  the  arithmetical  mean  of  the  two  results  obtained  for 
each  experiment,  is  given  in  column  7. 


EXPERIMENTS   ON   THE   FLOW  OF   WATER  OVER   WEIRS. 

Notwithstanding  the  advantage  attending  this  mode  of  reduction,  it  cannot  be 
denied  that,  for  the  most  perfect  experiments,  the  depth  on  the  weir  should  be 
invariable  throughout,  and  that,  cceteris  paribus,  the  experiments  will  be  the  less 
valuable,  the  greater  the  variation.  To  enable  the  reader  to  judge  of  the  rela- 
tive value  of  the  experiments,  as  far  as  it  depends  upon  this  variation,  the  small 
figures  to  the  left  and  above  the  several  depths  in  column  7  are  given;  they 
indicate  the  highest  number  of  values  of  h,  h',  h",  etc.  used  in  the  reduction  of 
the  observations,  at  either  of  the  hook  gauges,  in  the  corresponding  experiments. 

151.  COLUMN  8.     Mean  velocity  of  tlie  water  approaching  the  weir.     This  is  obtained 
by   dividing   the   corresponding   quantity   of  water   flowing   over  the   weir,   given   in 
column   14,  by   the   area   of   the   section   of    the   canal,   at   the   hook    gauge   boxes. 
In   the   weir    having    contraction    at   the    ends,   this^would    strictly   include    all    the 
space   under  the   gauge   boxes,  although,  from   the   form   of  the  walls,  it   is   evident 
that   the   current   could    flow   only   in    a    small    part   of    this   space  ;    consequently, 
the   portion   in   which   the   current   could   not   flow    is    not    included    in    the    areas 
used. 

152.  COLUMN  9.     Head   due    to    the    velocity    in    column    8.      This    is    sufficiently 
explained  in  the  heading. 

153.  COLUMN    10.     Depth    upon    the    weir,    corrected  for    the    velocity    of    the    water 
approaching  the  weir.     In  the  common  formula   for  the  discharge  of  water  over  weirs, 


The  second  member  may  be  separated  into  three  factors,  namely:  C,  the  coefficient 
of  contraction;  I,  the  length  of  the  weir;  and  ffft^ZglT,  the  theoretical  discharge 
for  the  unit  of  length.  According  to  a  well-known  elementary  theorem  in  hydrau- 
lics, the  latter  factor  may  be  represented  by  the  area  of  a  segment  of  a  parab- 
ola, of  which  the  parameter  is  2y;  thus,  in  figure  5,  plate  XII.,  if  AB  =  H, 
and  BC=\j2gir,  and  the  curve  A  MO  is  a  parabola,  of  which  the  vertex  is  A, 


we  shall  have  the  area  of  the  segment  AS  C^H^ZgH;  also,  the  velocity  of 
the  fluid  at  any  point  P  will  be  represented  by  the  ordinate  PM.  The  factor 
Hl^ZgH  may  also  be  decomposed  into  two  others:  H=AJB,  and  WZgH,  which 
equals  the  mean  value  of  all  the  ordinates  of  the  parabola  between  A  and  C, 
and  represents  the  mean  velocity  of  the  fluid  for  the  whole  height  of  the  ori- 
fice. In  demonstrating  this  theorem,  it  is  assumed  that  the  water  in  the  reser- 
voir is  at  rest  ;  we  can,  however,  easily  establish  an  analogous  theorem,  in  which 
it  is  assumed  that  the  water  in  the  reservoir  has  a  velocity  approaching  the 
weir,  in  the  direction  perpendicular  to  the  plane  of  the  weir.  Suppose  h  to  be 


EXPERIMENTS   ON  THE   FLOW   OF   WATER   OVER  WEIRS. 

the   head   due   this  velocity;    and  in  figure  6,  plate  XII.,  let  AB  —  H,  and  AD=: h, 
we   shall   have   for  the   velocity   v',  at   any   point   P   in   the   height  of  the   orifice, 


but   this   value    of  v'  is  the   ordinate    corresponding   to    the   abscissa,  AP-\-h  =  DP, 
of  a   parabola   whose   parameter  is   2^.     We   have    also 


We  can,  consequently,  represent  the  discharge  for  the  unit  of  length,  by  the  area 
of  the  surface  ABCG,  which  is  a  portion  of  the  segment  BCD;  the  area  of 
ABCG  is  the  difference  of  the  areas  of  the  segments  BOD  and  AGD;  the 
area  of  B  CD  is  • 


IBD  X  B  0=$(H+h)\/2ff(ir+h), 
and   the   area   of  ADG  is 


consequently,  the  area  of  ABCG  is 


and   for   the   total   discharge   we   have 


The   formula   (^4)    may   be    put   under   the   form 

Q=    Cll^TgH1. 

Suppose  H'  to   represent   a   depth   upon   the   weir    that   would    give    the   discharge 
Qf  by   the   formula   (  C  ),  we   shall   have 


substituting   the   value   of  Q'   in    (B~),   and    reducing,   we   find 


118  EXPERIMENTS   ON   THE   FLOW  OF   WATER  OVER  WEIRS. 

The  equation  (B),  from  which  this  value  of  H'  is  derived,  does  not  agree 
with  that  given  for  a  similar  case  by  most  writers  on  hydraulics,  who  seem 
generally  to  have  followed  Du  Buat  ;  *  it  agrees,  however,  with  the  expression 
given  by  Weisbach,f  who  appears  to  have  been  the  first  to  point  out  the  error. 

The  formula  (D)  was  communicated  to  the  author,  in  1849,  by  Mr.  Boyden, 
accompanied  by  a  demonstration  somewhat  resembling  the  above. 

The  values  of  IT,  given  in  column  10,  have  been  computed  by  the  formula 
(D)  from  the  corresponding  values  of  H  and  h  in  columns  7  and  9. 

154.  COLUMNS    11,    12,   and   13   are   sufficiently   explained    by  their    respective 
headings. 

155.  COLUMN  14.      Quantity   of  water  passing   the   weir  per  second.     The    quantities 
in    this   column    are    obtained    by   dividing    the   total    quantities    given    in   column 
13,  by  the  corresponding  intervals  of  time  in  column  6. 

156.  COLUMN  15.     Value  of  C  in  the  formula 


Q  having  the   corresponding   values   in   column   14. 

In   the   formula  proposed   at  art.  124,  namely:  — 

Q=  C(l—lnh)h*, 

the  values  of  the  constants  a  and  b  are  to  be  determined  by  experiment.  The 
values  adopted  in  the  formula  by  which  the  coefficients  in  this  column  have 
been  computed,  namely  :  a  =  f  ,  b  =  0.1,  were  determined  upon  after  many  trials 
of  other  values;  in  consequence  of  their  giving  results  according  the  most  nearly 
with  all  the  experiments,  and  at  the  same  time  having  a  convenient  degree  of 
simplicity.  It  is  quite  likely  that  many  other  values  of  a  and  b  (probably  an 
unlimited  number)  might  be  found  that  would  accord  somewhat  nearer  with  the 
experiments;  a  closer  approximation  than  is  given  by  the  use  of  the  values 
adopted,  could  have,  however,  but  little  practical  value  ;  much  less,  it  was  thought, 
than  would  be  derived  from  the  use  of  the  simple  values  adopted.  The  use  of 
a  fractional  power,  such  as  a  =1.47,  deduced  from  the  experiments  at  the  Tre- 
mont  Turbine  (art.  135),  is  very  inconvenient,  and,  to  persons  not  well  skilled  in 
the  use  of  logarithms,  offers  great  difficulty. 


*  Principes  cT  Hydraulique,  etc.,  by  M.  Du  Buat.     Paris:  1816.  Vol.  1,  page  201. 
t  Attgemeine  Maschinen  Encyclopiidie.     Leipzig :  1841.  Vol.  1,  page  489. 


EXPERIMENTS   ON   THE   FLOW   OF   WATER  OVER  WEIRS. 

157.  COLUMNS   16,  17,  and    18,  are,  for   the   purpose   of  obtaining   correct   mean 
results   of  the    experiments,   made   under   circumstances   nearly   identical.      In   conse- 
quence  of   the   variations   in   the   height   of    the    canal   (art.    150),   it   was   impracti- 
cable  to   repeat   the    experiments  with    precisely   the    same    depth   upon   the   weir  ; 
by   the   method    adopted   for   obtaining    these   mean   results,   all    inconvenience   from 
this   source   is   obviated.      As    the    formula    by   which   the   values   of    C,   in   column 
15,  are   obtained,  is   such   as   to   give  results  agreeing  very  nearly  with  experiment, 
even  when   the  depths  differ  considerably,  it  is  plain  that  the  values  of  C  deduced 
from   experiments    having    nearly   the    same    depths,   cannot    be    affected    by   small 
variations   in   the    depths,   and   will   be    subject   to   no    greater  irregularities   than   if, 
in   the   several   experiments   from   which    they   are    deduced,    the    depths    had    been 
precisely   the   same.     We    can   consequently  take   a   mean    coefficient  with   the   same 
confidence   that  we  could   take   a   mean  quantity,  if  the  depths   had   been   precisely 
the   same.      These   mean   coefficients   are   given    in   column    16.      In   column    17   are 
given   depths   on   the    weir,  nearly  a  mean    of  those   in  the  experiments  from  which 
the   corresponding   mean   coefficients   have  been  deduced.      In  column    18,  are  given 
what   may  be  called  the  mean   quantities  of  water  actually  found  by  experiment  to 
be  discharged  with   the   corresponding   depths  in   column    17.      A  method    similar  to 
the   above   was    used   to   reduce    the    quantities    discharged    in    the    experiments    of 
Castel,    reported    in    the    Aniiales    de    chimie    et    de    Physique,   vol.    62.   Paris:    1836; 
reprinted   in   the   first   volume    of  the  Annales  des  Pords  et   Chaussees  for    1837. 

158.  COLUMN  19.     Quantity  of  water  passing  the  weir,  calculated  by  the  formula 


H"  having  the  corresponding  values  in  column  17. 

The  coefficient  3.33  is  derived  from  the  arithmetical  mean  of  all  the  coeffi- 
cients in  column  15,  which  is  3.3318,  the  two  final  decimals  being  omitted  for 
the  sake  of  simplicity.  The  largest  coefficient  in  column  15,  is  that  deduced 
from  experiment  34,  which  is  3.3617,  exceeding  the  coefficient  adopted  by  T|^ 
part;  the  smallest  coefficient  is  that  deduced  from  experiment  4,  which  is  3.3002, 
being  less  than  the  coefficient  adopted,  by  ^^  part;  that  is,  the  formula  by 
which  the  quantities  in  column  19  are  computed,  will  represent  every  experi- 
ment in  the  table,  within  one  per  cent. 

159.  COLUMN  20.  Proportional  difference,  or  the  absolute  difference  of  the  quantities 
in  columns  18  and  19,  divided  by  the  quantity  in  column  18.  The  greatest  propor- 
tional difference  is  that  deduced  from  experiments  34  and  35,  which  is  —  0.0090, 
or  a  little  less  than  one  per  cent.  In  these  experiments  there  were  two  weirs, 


120  EXPERIMENTS  ON  THE   FLOW   OF   WATER  OVER   WEIRS. 

about  four  feet  long  each,  separated  by  a  partition  two  feet  wide ;  the  near 
neighborhood  of  the  two  orifices  appears  to  have  affected  the  discharge.  The 
next  largest  proportional  difference  is  that  deduced  from  experiments  36  to 
43,  which  is  — 0.0068,  or  about  f  of  one  per  cent. ;  in  these  experiments,  the 
depth  of  the  water  in  the  canal  leading  to  the  weir,  was  only  about  three  times 
the  depth  upon  the  weir.  The  experiments  with  the  diminished  depth  in  the 
canal  were  made  for  the  purpose  of  testing  the  method  of  correcting  the  depths, 
upon  the  weir,  for  the  velocity  of  the  water  approaching  the  weir  (art.  153). 
They  indicate  that  the  method  is  not  strictly  accurate,  as  might  have  been 
anticipated,  omitting,  as  it  does,  all  consideration  of  the  effect  produced  by  this 
velocity,  in  modifying  the  contraction.  It  is  well  understood  that  such  an  effect 
is  produced,*  but  it  is  of  such  a  complicated  nature,  that  the  investigations  hith- 
erto undertaken  have  thrown  but  little  light  upon  it. 

It  will  be  perceived  by  referring  to  column  4,  that  the  experiments  51  to 
55  were  made  under  the  same  circumstances  as  experiments  44  to  50,  excepting 
that  the  sheet  of  water,  after  passing  the  weir,  was  prevented  from  expanding 
laterally  for  a  certain  distance.  This  was  accomplished  by  placing  boards  at  the 
ends  of  the  sheet,  as  represented  by  the  broken  lines  at  A,  figures  8  and  9, 
plate  XIV.  By  referring  to  column  16,  it  will  be  seen  that  the  effect  of  these 
boards  was  to  diminish  the  coefficient  from  3.3409  to  3.3270,  corresponding  to  a 
diminution  of  the  quantity  discharged  by  the  weir,  with  the  same  depth,  of  •%%•$, 
or  about  four-tenths  of  one  per  cent. ;  in  other  words,  the  effect  of  the  boards 
upon  the  discharge  was  the  same  as  would  be  produced  by  shortening  the  weir 
?ITP  or  \  inch,  at  each  end.  By  reference  to  figure  8,  plate  XIV.,  it  will  be 
perceived  that  these  boards  did  not  affect  the  free  communication  between  the 
atmosphere  and  the  air  under  the  sheet  of  water ;  if  this  communication  had 
been  obstructed,  so  that  the  pressure  of  the  air  under  the  sheet  had  been  dif- 
ferent from  that  of  the  atmosphere,  it  would  have  affected  the  discharge. 

*  Jaugeage  des  cours  d'eau,  etc.,  by  M.  P.  Boikau,  page  40.     Paris:  1850. 


122 


EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS. 


TABLE 

EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS,  MADE  AT  THE 


1 

a 

3 

4 

5 

6 

7 

8 

_ 

Mean 

Temperatures  by 

velocity  of 

Fahrenheit's 

the  water 

thermometer. 

Tune  of  the  commencement  and 
conclusion  of  the  experiment, 

approach- 
ing the 

Num 
her  of 

Date  of  the 

Reference  to  the  figures  on  plate  XIV.,  and  particular 

as  indicated  by  the 
telegraphic  signals. 

Dura- 
tion of 
the 

Mean  depth 
upon  the 
weir  by 

weir  at  the 
transverse 
sect,  thro' 
the  holes 
in  the 

the 
expert 

experiment. 

description  of  the  weir. 

experi- 

observation. 

hook 
gauge 

ment. 

1852. 

Of  the  all 

Of  the 

ment. 

boxes,  or 

in  the 
shade. 

water. 

Commencement. 

Conclusion. 

H. 

six  feet 
from  the 
weir. 

V. 

H. 

niiti 

sec. 

H. 

tnin 

sec. 

we. 

Feet. 

Feet. 

1 

Oct.  27,  P.M 

Figures  1,  2,  and  3. 

10 

15 

0.8 

10 

18 

13.6 

192.8 

21.52430 

0.7682 

2 
3 

a      it      tt 
tt      a      tt 

Width  of  the  canal  on  the  upstream  side  of  the  weir, 
13.96  feet.    Mean  depth  of  the  canal  opposite  the  hook 
gauge  boxes  6.048  feet  below  the  top  of  the  weir. 

11 
11 

20 
54 

1.3 
1.3 

•11 
11 

23 
57 

14.2 
10.6 

192.9 
189.3 

21.55045 
21.55930 

0.7813 

0.7882 

4 

"      28,  A.M 

43.75° 

46.5° 

0 

26 

17.9 

0 

29 

20.1 

182.2 

'i.segio 

0.7889 

5 

Oct.  24,  P.M 

52° 

48.5° 

f 

i 

0.5 

c 

8 

20.5 

260.0 

"1.23690 

0.5904 

6 

It          U          it 

C 

33 

2.3 

( 

37 

21.5 

259.2 

21.24195 

0.5933 

7 

It          ft          It 

Figures  1,  2,  and  8. 

10 

0 

1.7 

10 

4 

18.4 

256.7 

21.24795 

0.5971 

8 

11          U          tt 

Same  as  the  preceding. 

10 

31 

2.1 

10 

35 

20.6 

258.5 

21.25085 

0.5944 

9 

tl          ft          U 

11 

0 

2.3 

11 

4 

16.1 

253.8 

"1.25290 

0.6000 

10 

U          tl          tt 

11 

30 

1.8 

11 

34 

21.4 

259.6 

21.25490 

0.5987 

11 

Oct.  20,  P.M 

10 

1 

0.8 

10 

7 

22.0 

381.2 

"0.96711 

0.4256 

12 

U          U          tl 

. 

10 

30 

0.9 

10 

.'So 

44.0 

343.1 

"1.02755 

0.4594 

13 

ft        tl        It 

11 

12 

0.7 

11 

17 

47.4 

346.7 

"1.03395 

0.4629 

14 

tl        11        il 

11 

48 

0.6 

11 

53 

43.3 

342.7 

21.03315 

0.4634 

15 

"      21,  A.  M. 

0 

24 

59.5 

0 

30 

37.5 

338.0 

"1.04060 

0.4680 

16 

11          It          U 

43° 

49° 

1 

0 

0.0 

1 

5 

37.4 

337.4 

"1.03735 

0.4666 

17 

"       "     P.M. 

c 

48 

8.0 

c 

54 

36.4 

388.4 

"0.96325 

0.4233 

18 

it       u       ft 

10 

23 

1.2 

10 

29 

9.9 

368.7 

"0.97590 

0.4304 

19 

it       u       tt 

42° 

48.75° 

10 

52 

0.4 

10 

58 

8.7 

368.3 

"0.97950 

0.4318 

20 

It       ft       11 

11 

23 

1.4 

11 

29 

4.2 

362.8 

"0.98885 

0.4377 

21 

It        tl        tl 

Figures  1.  2,  and  3. 

11 

53 

1.5 

11 

59 

3.5 

362.0 

"0.99460 

0.4418 

22 

"      22,  A.  M. 

Same  as  the  preceding. 

0 

43 

0.0 

0 

49 

48.8 

408.8 

"0.91570 

0.3951 

23 

11          ft          11 

42° 

1 

12 

0.7 

1 

18 

40.9 

400.2 

20.92800 

0.4015 

24 

u       u       a 

1 

42 

0.3 

1 

48 

26.8 

386.5 

20.94625 

0.4126 

25 

"      29,  P.M. 

9 

2 

3.5 

9 

7 

54.2 

350.7 

21.01275 

0.4517 

26 

It          It          U 

51.5° 

48.25° 

9 

35 

2.7 

9 

40 

51.6 

348.9 

41.01160 

0.4520 

27 

It          tl          U 

10 

5 

1.2 

10 

10 

59.7 

358.5 

00.99495 

0.4429 

28 

U          U          11 

10 

34 

59.8 

10 

40 

39.7 

339.9 

n.03360 

0.4653 

29 

il          it          H 

11 

3 

0.2 

11 

8 

29.6 

329.4 

"1.05565 

0.4779 

30 

U          il          il 

11 

32 

1.8 

11 

37 

27.0 

325.2 

21.06920 

0.4863 

31 

Nov.  11,  P.M. 

34° 

41.25° 

8 

56 

59.5 

9 

3 

6.0 

366.5 

20.98370 

0.4352 

32 

u      u      u 

9 

30 

0.6 

9 

36 

7.8 

367.2 

"0.97820 

0.4320 

33 

u       u       u 

10 

0 

0.3 

10 

6 

19.8 

379.5 

*0.96700 

0.4236 

Figure  4. 

34 
35 

Nov.  3,  P.M. 

«       il       it 

45° 

48° 

Width  of  the  canal  on  the  upstream  side  of  the  weir, 
18.%  feet.     Mean  depth  of  the  canal  opposite  the  hook 
gauge  boxes,  6.048  feet  below  top  of  the  wet.     Two 

9 
9 

12 

59 

1.3 

59.6 

9 

10 

19 

7 

37.8 
18.4 

456.5 
438.8 

61.01025 
61.02625 

0.3527 
0.3596 

equal  bays  separated  by  a  partition  2  feet  wide. 

36 

Oct.   31,  A.M. 

- 

7 

17 

15.8 

7 

22 

52.9 

337.1 

"1.02805 

0.9496 

37 

It        it        tl 

7 

47 

59.6 

7 

53 

29.9 

330.3 

a1.03720 

0.9589 

38 

u       u       a 

46° 

48.75° 

Figures  6,  6,  and  7. 

9 

46 

0.3 

9 

51 

34.9 

334.6 

"1.04455 

0.9684 

39 
40 
41 

u       u       a 
it       it       it 
u       u       u 

Width  of  the  canal  on  the  upstream  side  of  the  weir, 
18.96  feet.     Mean  depth  of  the  canal  opposite  the  hook 
gauge  boxes,  2.014  feet  below  the  top  of  the  weir.    Bot- 
tom of  the  canal  horizontal  for  23  feet  on  the  upstream 
side  of  the  weir. 

10 
10 
11 

14 
41 
10 

1.4 
0.7 

7.8 

10 
10 
11 

19 
46 
15 

23.3 
28.2 
32.5 

321.9 
327.5 
324.7 

21.  04495 
21.04600 
51.05130 

0.9693 
0.9691 
0.9756 

42 

li          11          11 

11 

39 

59.4 

11 

45 

8.3 

308.9 

"1.07945 

1.0049 

43 

"       "     P.M. 

0 

15 

1.4 

0 

20 

15.7 

314.3 

21.07115 

0.9958 

EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS. 


123 


XIII. 

LOWER  LOCKS,   LOWELL,   IN   OCTOBER  AND   NOVEMBER,  1852. 


9 

10 

11 

IS 

13 

14 

15 

16 

17 

18 

19 

ao 

Depth  upon  the 

Quantity  of  water 

Head  due  to 

weir,  corrected 

Total  quan- 

Approx- 

that would  have 

Proportional 

Num- 
ber of 
the 
experi- 
ment. 

the  Telocity 
n  column  8, 
or  the 
values  of 
k  by  the 
formula 

A-      + 

for  the  velocity 
of  the  water  ap- 
proaching the 
weir,  or  the  val- 
ues of  H'  by  the 
formula 
H'= 
£      32 

Length 
of  the 
weir 
I. 

No. 
of 
end 
con- 
trac- 
tions. 
n. 

tity  of  water 
that  passed 
the  weir  dur- 
ing each  ex- 
periment, as 
measured  in 
the  lock 
chamber. 

Quantity  of 
water  pass- 
ing the  weir 
per  second. 

Value  of  C  in 
the  formula 

C(l—0.1nH')H'% 
Q  having  the 
corresponding 
values  in  col- 
umn 14. 

Mean 
value  of 
C  for  each 
particular 
descrip- 
tion of 
weir. 

imate 
mean 
depth 
upon 
the  weir, 
for  each 
particu- 
lar de- 
scription 
of  weir. 

passed  the  weir 
with  the  depth  in 
column  17,  calcula- 
ted by  the  formula 

Q=            8 

qj—  0.1nH'/)H"2 
C  having  the  cor- 
responding value 
in  column  16. 

Quantity  of  water 
passing  the  weir, 
calculated  by  the 
formula 

Q=       % 

3.33(l-Q.lnIP>)H"* 

difference,  or 
the  absolute 
difference  of 
the  quantities 
in  columns  18 
and  19,  divided 
by  the  quan- 
tity in  column 

64.3236. 

[(H+A)5-«]3 

W. 

18. 

Feet. 

Feet. 

Feet. 

Cubic  feet. 

Cubic  feet 

Feet. 

Cubic  feet  per 

Cubic  feet  per 

per  second. 

second. 

second. 

1 

0.00917 

1.53300 

9.997 

2 

11815.19 

61.2821 

3.3318 

2 

0.00949 

1.55945 

u 

U 

12069.49 

62.5686 

3.3174 

3 

0.00966 

1.56845 

a 

tt 

11964.90 

63.2060 

3.3230 

3.3181 

1.56 

62.6147 

62.8392 

+  0.0036 

4 

0.00968 

1.57828 

it 

tt 

11542.52 

63.3508 

3.3002 

5 

0.00542 

1.24208 

9.997 

2 

11723.21 

45.0893 

3.3412 

6 

0.00547 

1.24718 

u 

tt 

11753.09 

45.3437 

3.3398 

7 

0.00554 

1.25325 

tt 

tt 

11725.57 

45.6781 

3.3405 

8 

0.00549 

1.25610 

it 

tt 

11760.23 

45.4941 

3.3159 

3.3338 

1.25 

45.4125 

45.3608 

—  0.0011 

9 

0.00560 

1.25825 

a 

tt 

11658.13 

45.9343 

3.3396 

-> 

10 

0.00557 

1.26022 

tt 

tt 

11903.41 

45.8529 

3.3260 

11 

0.00282 

0.96983 

9.997 

2 

11872.75 

31.1457 

3.3265 

12 

0.00328 

1.03071 

U 

u 

11645.35 

33.9416 

3.3129 

13 

0.00333 

1.03716 

U 

u 

11869.83 

34.2366 

3.3110 

14 

0.00334 

1.03636 

tt 

It 

11745.20 

34.2725 

3.3182 

15 

0.00341 

1.04388 

u 

tt 

11713.35 

34.6549 

3.3196 

16 

0.00339 

1.04061 

tt 

tt 

11651.45 

34.5330 

3.3233 

17 

0.00279 

0.9  6593 

It 

tt 

12023.62 

30.9568 

3.3261 

18 

0.00288 

0.97868 

tt 

it 

11627.93 

31.5377 

3.3234 

19 

0.00290 

0.98229 

tt 

tt 

11659.61 

31.6579 

3.3179 

20 

0.00298 

0.99172 

it 

It 

11661.78 

32.1438 

3.3216 

21 

0.00303 

0.99752 

It 

tt 

11754.36 

32.4706 

3.3265 

3.3223 

1.00 

32.5486 

32.6240 

+  0.0023 

22 

0.00243 

0.91804 

tt 

tt 

11721.88 

28.6739 

3.3218 

23 

0.00251 

0.93042 

tt 

It 

11682.99 

29.1929 

3.3155 

24 

0.00265 

0.94881 

tt 

tt 

11629.93 

30.0904 

3.3198 

25 

0.00317 

1.01580 

tt 

tt 

11678.40 

33.3003 

3.3211 

26 

0.00318 

1.01466 

It 

tt 

11623.48 

33.3147 

3.3281 

27 

0.00305 

0.99789 

tt 

tt 

11670.68 

32.5542 

3.3333 

28 

0.00337 

1.03684 

tt 

tt 

11697.19 

34.4136 

3.3297 

29 

0.00355 

1.05906 

u 

tt 

11685.17 

35.4741 

3.3263 

30 

0.00368 

1.07274 

tt 

tt 

11764.18 

36.1752 

3.3283 

31 

0.00294 

0.98653 

tt 

tt 

11702.07 

31.9293 

3.3251 

32 

0.00290 

0.98099 

tt 

tt 

11629.66 

31.6712 

3.3259 

33 

0.00279 

0.96969 

tt 

tt 

11762.21 

30.9940 

3.3111 

34 

0.00193 

1.01212 

7.997 

4 

11863.64 

25.9883 

3.3617 

35 

0.00201 

1.02820 

U 

if 

11655.85 

26.5630 

3.3586 

3.3601 

1.02 

26.2686 

26.0333 

—  0.0090 

36' 

0.01402 

1.04098 

9.997 

2 

11747.38 

34.8484 

3.3519 

37 

0.01430 

1.05039 

U 

It 

11657.37 

35.2933 

3.3498 

38 

0.01458 

1.05799 

a 

it 

11953.56 

35.7249 

3.3548 

39 

0.01461 

1.05842 

tt 

tt 

11513.09 

35.7660 

3.3567 

40 

0.01460 

1.05946 

tt 

tt 

11715.10 

35.7713 

3.3523 

3.3527 

1.06 

35.8026 

35.5602 

—  0.0068 

41 

0.01480 

1.06494 

tt 

tt 

11712.43 

36.0716 

3.3548 

42 

0.01570 

1.09390 

tt 

tt 

11579.82 

37.4873 

3.3509 

43 

0.01542       1.08535 

tt 

tt 

11645.21  37.0513 

3.3505 

L 

124 


EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS. 


j  TABLE 

EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS,  MADE  AT  THE 


1 

3 

a 

4 

5 

6 

7 

8 

Mean 

Temperatures  by 

velocity  of 

Fahrenheit's 

Tune  of  the  commencement  and 

the'  water 
approach- 

thermometer. 

conclusion  of  the  experiment, 

ing  the 
weir  at  the 

Dura- 

Mean depth 

transverse 

Num 

telegraphic  signals. 

sect,  thro' 

tion  of 

upon  the 

the  holes 

ber  o: 

Date  of  the 

Reference  to  the  figures  on  plate  XIV.,  and  particular 

the 

weir  by 

in  the 

the 
experi 

experiment. 

description  of  the  weir. 

experi- 

observation. 

hook 
gauge 

1852. 

Of  the  ail 

Of  the 

ment. 

boxes,  or 

in  the 

water. 

Commencement. 

Conclusion. 

a. 

six  feet 
from  the 

shade 

weir. 

V. 

H. 

min 

sec. 

H. 

pun 

sec. 

nee. 

Feet. 

44 

Nov.  7,  A.M 

44° 

44° 

Figures  8,  9,  and  10. 

7 

50 

1.0 

7 

55 

51.2 

350.2 

"0.98675 

0.5455 

45 
46 

a      u      u 
ii      11      tt 

Mean  width  of  the  canal  for  20  feet  on  the  upstream 
side  of  the  weir,  9.992  feet.    Mean  depth  of  the  canal 
opposite  the  hook  gauge  boxes,  5.048  feet  below  the  top 

10 

38 
11 

0.0 

59.2 

j 
10 

43 
17 

53.7 
57.4 

353.7 
358.2 

40.98490 
"0.97450 

0.5446 
0.5376 

47 

11          U          11 

of  the  weir. 

10 

43 

59.6 

10 

49 

59.0 

359.4 

"0.97620 

0.5385 

48 

U          It          tl 

11 

15 

0.0 

11 

21 

0.4 

360.4 

"0.97600 

0.5387 

49 

U          11          11 

11 

48 

4.4 

11 

54 

0.7 

356.3 

40.97775 

0.5394 

50 

"       "     P.M. 

0 

21 

0.2 

0 

26 

58.3 

358.1 

"0.97690 

0.5390 

51 

Nov.  7,  P.M. 

42.25° 

43.75° 

Figures  8,  9,  and  10. 

8 

23 

7.6 

8 

28 

52.0 

344.4 

81.00505 

0.5589 

52 

11          11          U 

Width  and  depth  same  as  the  preceding.    The  sheet 

8 

55 

59.7 

9 

1 

46.9 

347.2 

21.00600 

0.5581 

53 

11        11        11 

of  water  after  passing  the  weir,  was  prevented  from 
expanding  in  widthj  for  a  certain  distance,  by  boards  at 

q 

B 

28 

3.1 

9 

33 

51.3 

348.2 

"1.00520 

0.5574 

54 

11        It        11 

each  end  of  the  weir,  placed  in  the  same  planes  as  the 

c 

59 

59.8 

10 

5 

52.9 

353.1 

40.99265 

0.5480 

55 

U          tl          tl 

sides  of  the  canal  leading  to  the  weir. 

10 

31 

1.5 

10 

36 

54.0 

352.5 

"0.99240 

0.5477 

56 

Oct.  24,  P.M. 

Figures  1,  2,  and  8. 

~2 

24 

59.3 

2 

32 

53.8 

474.5 

*0.81860 

0.3405 

57 

U          H          u 

Width  of  the  canal  on  '.he  upstream  side  of  the  wen-, 

3 

3 

0.4 

3 

11 

11.1 

490.7 

^.80755 

0.3338 

58 
59 

11           11           U 
tl          U          11 

13.96  feet.     Mean  depth  of  the  canal  opposite  the  hook 
gauge  boxes,  5.048  ieet  below  the  top  of  the  wea. 

3 
4 

40 
17 

0.3 
4.7 

3 

4 

48 
25 

19.9 
44.9 

499.6 
520.2 

"0.77690 

0.3272 
0.3170 

60 

It       11       11 

4 

55 

1.7 

5 

3 

18.6 

496.9 

"0.80125 

0.3306 

61 

11          11          U 

5 

29 

1.8 

5 

37 

27.6 

505.8 

"0.79400 

0.3258 

62 

Oct.  31,  P.M. 

Figures  5,  6,  and  7. 

~2 

20 

0.3 

2 

28 

40.5 

520.2 

"0.77115 

0.6694 

63 

U           tl           tl 

Width  of  the  canal  on  the  upstream  side  of  the  weir, 

3 

0 

1.3 

3 

8 

32.4 

511.1 

'0.78725 

0.6872 

64 
65 

U           tt            tt 

It        It        It 

47° 

48.75° 

18.96  feet.     Mean  depth  of  the  canal  opposite  the  hook 
gauge  boxes,  2.014  feet  below  the  top  of  the  weir.    Bot- 
tom of  the  canal  horizontal,  for  23  feet  on  the  upstream 

3 
4 

38 
14 

4.4 
0.2 

3 
4 

46 
21 

7.9 
5.5 

483.5 
425.3 

"0.80455 
"0.87960 

0.7052 
0.7870 

66 

11            U           tl 

side  of  the  weir. 

4 

47 

58.0 

4 

54 

56.2 

418.2 

"0.88865 

0.7963 

67 

Nov.  7,  P.M. 

Figures  8,  9,  and  10. 

2 

7 

2.7 

2 

16 

14.7 

552.0 

"0.73620 

0.3659 

68 

11        tl        It 

Mean  width  of  the  canal  for  20  feet  on  the  upstream 

2 

43 

1.0 

2 

51 

6.8 

485.8 

*0.80195 

0.4122 

69 

It        tt        11 

side  of  the  weir,  9.992  feet.     Mean  depth  of  ibe  canal 
opposite  the  hook  gauge  boxes,  5.048  feet  below  the  top 

3 

17 

59.7 

3 

25 

56.0 

476.3 

"0.80950 

0.4176 

70 

It        11        tl 

of  the  weir. 

3 

51 

59.9 

3 

59 

51.7 

471.8 

*O.SU95 

0.4213 

71 

tl        11        11 

4 

25 

0.0 

4 

32 

53.9 

473.9 

'0.81325 

0.4192 

72 

Oct.  24,  A.M. 

Figures  1,  2,  and  3. 

7 

12 

2.5 

7 

25 

9.4 

786.9 

"0.59190 

0.2182 

73 

11        It        tt 

46.5° 

47.75° 

Width  of  the  canal  on  the  upstream  side  of  the  weir, 

7 

49 

59.8 

8 

2 

52.7 

772.9 

"0.59240 

0.2186 

74 

It       tl       tt 

13.96  feet.     Mean  depth  of  the  canal  opposite  the  hook 
gauge  boxes  5.048  feet  below  the  top  of  the  weir. 

9 

11 

59.1 

9 

24 

36.6 

757.5 

"0.61060 

0.2279 

75 

It        It        tl 

10 

33 

59.7 

10 

45 

14.4 

674.7 

"0.65525 

0.2509 

76 

It        It        It 

59.5° 

48° 

11 

8 

1.4 

11 

19 

27.9 

686.5 

60.64305 

0.2449 

77 

11       It       tl 

11 

50 

0.3 

12 

1 

35.3 

695.0 

40.63795 

0.2419 

78 

«      "    P.M.  64.5° 

48.5° 

0 

24 

58.5 

0 

36 

42.9 

704.4 

"0.63370 

0.2396 

79 

Oct.  31,  P.M. 

Figures  5,  6,  and  7. 

~7 

~6 

59.6 

7 

18 

8.6 

669.0 

"0.65150 

0.5405 

80 

u      n      it 

Width  of  the  canal  on  the  upstream  side  of  the  weir, 

7 

46 

0.3 

7 

57 

9.3 

669.0 

"0.65590 

0.5455 

81 

it      u      tt 

45.25° 

48.75° 

13.96  feet.     Mean  depth  of  the  canal  opposite  the  hook 
gauge  boxes,  2.014  feet  below  the  top  of  the  weir.    Bot- 

8 

24 

0.0 

8 

34 

56.9 

656.9 

"0.65985 

0.5496 

82 

u      it      11 

tom  of  the  canal  horizontal  for  23  feet  on  the  upstream 

9 

0 

59.4 

9 

12 

42.4 

703.0 

40.63135 

0.5193 

83 

it      it      n 

side  of  the  weir. 

9 

40 

1.0 

9 

51 

27.8 

686.8 

"0.64250 

0.5309 

84 

n      u      n 

10 

23 

1.7 

10 

34 

11.0 

669.3 

"0.65460 

0.5439 

85 

Oct.  31,  P.M. 

Figures  5,  6,  and  4. 

11 

43 

0.7 

11 

56 

42.8 

822.1 

60.66940 

0.4382 

86 

87 

NOV.    1,    A.M. 

tt            It           H 

Width,  depth,  and  bottom  of  the  canal  same  as  the 
preceding.     Two  equal  bays  separated  by  a  partition  2 
feet  wide. 

0 
1 

21 
0 

59.9 
3.8 

0 

1 

35 
13 

26.0 
17.9 

806.1 
794.1 

"0.67900 
"0.68360 

0.4459 
0.4496 

«8 

It           11           11 

1 

39 

59.8 

1 

53 

9.5 

789.7 

"0.68815 

0.4526 

EXPEEIMENTS   ON  THE   FLOW  OF   WATER  OVER   WEIRS. 


125 


X  1 1 1  —  CONTINUED. 


LOWER   LOCKS,  LOWELL,   IN   OCTOBER  AND   NOVEMBER,   1852. 


1  

9 

.10 

11 

12 

13 

14 

15 

1G 

17 

18 

19 

20 

Depth  upon  the 

Quantity  of  water 

Head  due  tc 

weir  corrected 

Total  quan- 

Appro* 

that  would  have 

Proportional 

Num- 
ber of 
the 

experi- 
ment. 

the  yelocit1 
in  column  8 
or  the 
values  of 
h  by  the 
formula 

A-      7" 

1  for  the  velocity 
of  the  water  ap- 
proaching the 
weir,  or  the  val- 
ues of  Hf  by  the 
formula 
#'= 

Length 
of  the 
weir. 
1. 

No. 
of 
end 
con- 
trac- 
tions 
n. 

tity  of  water 
that  passed 
the  weir  dur- 
ing each  ex- 
periment, as 
measured  in 
the  lock 

Quantity  of 
water  pass- 
ing the  well 
per  second. 

Value  of  C  in 
the  formula 

Q=        5 

C(l—  0.1«H')H/2 
Q  having  the 
corresponding 
values  in  col- 

Mean 
value  of 
C  for  each 
particular 
descrip- 
tion of 
weir. 

imatc* 
mean 
depth 
upon 
the  weir 
for  each 
particu- 
lar de- 
scription 
of  weir 

passed  the  weir 
with  the  depth  in 
column  17,  calcula- 
ted by  the  formula. 

Q=            8 

qj-o.inH"jw2 

C  having  the  cor- 
responding value 
in  column  16. 

Quantity  of  water 
passing  the  weir, 
calculated  by  the 
formula 

Q=            J 

3.33(Mnn.H")/f''5 

difference,  or 
the  absolute 
difference  of 
the  quantities 
in  columns  1£ 
and  19,  divided 
by  the  quan- 

64.323b 

.  ,  i      48 

chamber. 

umn  14. 

tity  in  column 

[(H+/ift—  A5J3 

• 

Sf'. 

18. 

Feet. 

Feet. 

Feet. 

Cubic  feet. 

Cubic  feet. 

Feet. 

Cubic  feet  per 

Cubic  feet  per 

second. 

second. 

44 

0.00463 

0.99117 

9.995 

0 

11524.62 

32.9087 

3.3366 

45 

0.0046 

0.98930 

u 

tt 

11616.54 

32.8429 

3.3394 

46 

0.00449 

0.97879 

tt 

tt 

11592.18  32.3623 

3.3437 

47 

0.00451 

0.98051 

it 

tt 

11655.28  32.4299 

3.3418 

3.3409 

0.98 

32.3956 

32.2899 

—  0.0033 

48 

0.0045 

0.98031 

tt 

tt 

11689.79  32.4356 

3.3434 

49 

0.00452 

0.98207 

it 

it 

11576.77 

32.4916 

3.3402 

50 

0.00452 

0.98122 

it 

tt 

11623.93 

32.4600 

3.3413 

51 

0.00480 

1.00968 

9.995 

0 

11646.88 

33.8179 

3.3349 

52 

0.00484 

1.01061 

tt 

tt 

11725.23 

33.7708 

3.3257 

53 

0.00483 

1.00980 

tt 

tt 

11743.85 

33.7273 

3.3254 

3.3270 

1.00 

33.2534 

33.2833 

+  0.0009 

54 

0.00467 

0.99710 

U 

tt 

11683.48  33.0883 

3.3249 

55 

0'.00466 

0.99684 

it 

tt 

11656.76 

33.0688 

3.3243 

56 

0.0018C 

0.82034 

9.997,    2 

11539.45  24.3192 

3.3287 

57 

0.00173 

0.80923 

U 

tt 

11675.86  23.7943 

3.3234 

58 

0.00166 

0.79726 

tt 

tt 

11628.76  23.2761 

3.3237 

3.3246 

0.80 

23.4011 

23.4391 

+  0.0016 

59 

0.00156 

0.77842 

It 

tt 

11694.31   22.4804 

3.3261 

60 

0.00170 

0.80290 

U 

tt 

11698.38 

23.5427 

3.3268 

61 

0.00165 

0.79560 

M 

tt 

11719.40 

23.1700 

3.3188 

62 

0.00697 

0.77768 

9.997 

~2 

11718.53 

22.5270 

3.3376 

63 

0.00734 

0.79412 

it 

tt 

11887.02 

23.2577 

3.3406 

64 

0.00773 

0.81178 

U 

tt 

11610.17 

24.0128 

3.3383 

3.3403 

Q.83 

24.8313 

24.7548 

—  0.0031 

65 

0.00963 

0.88855 

it 

tt 

11695.06 

27.4984 

3.3435 

66 

0.00986 

0.89782 

It 

tt 

11671.63 

27.9092 

3.3417 

67 

0.00208 

0.73821 

9.995 

0 

11676.57 

21.1532 

3.3368 

68 

0.00264 

0.80449 

tt 

a 

11709.76 

24.1041 

3.3422 

69 

0.00271 

0.81211 

tt 

tt 

11645.16 

24.4492 

3.3424 

3.3393 

0.80 

23.8821 

23.8156 

—  0.0028 

70 

0.00276 

0.81760 

it 

tt 

11647.58 

24.6876 

3.3410 

71 

0.00273 

0.81588 

It 

tt 

11638.23 

24.5584 

3.3341 

72 

0.00074 

0.59262 

9.997 

2 

11803.57 

15.0001 

3.3284 

73 

0.00074 

0.59312 

tt 

tt 

11614.60 

15.0273 

3.3303 

74 

0.00081 

0.61139 

tt 

tt 

11902.13 

15.7124 

3.3284 

75 

0.00098 

0.65621 

tt 

tt 

11760.381 

17.4305 

3.3237 

3.3275 

0.62 

16.0382 

16.0502 

+  0.0008 

76 

0.00093 

0.64395 

tt 

tt 

11659.42 

16.9839 

3.3306 

77 

0.00091 

0.63883 

tt 

It 

11648.02 

16.7597 

3.3259 

78 

0.00089 

0.63456 

tt 

tt 

1  1  685.54 

16.5894 

3.3250 

79 

0.00454 

0.65579 

9.997 

2 

11657.19 

17.4248 

3.3258 

80 

0.00463 

0.66027 

u 

It 

11783.30 

17.6133 

3.3278 

81 

0.00470 

0.66429 

tt 

tt 

11674.77 

17.7725 

3.3278 

3.3262 

0.65 

17.1990 

17.2187 

+  0.0011 

82 

0.00419 

0.63532 

tt 

tt 

1  1  682.49 

16.6181 

3.3249 

83 

0.00438 

0.64664 

tt 

tt 

11715.28 

17.0578 

3.3244 

84 

0.00460 

0.65894 

tt 

tt 

11748.86 

17.5540 

3.3266 

85 

0.00299 

0.67226 

7.997 

4 

11690.02 

14.2197 

3.3382 

86 

0,00309 

0.68195 

It 

tt 

11703.92 

14.5192 

3.3378 

87 

0.00314 

0.68660 

tt 

tt 

11644.82 

14.6642 

3.3378 

3.3368 

0.68 

14.4541 

14.4247 

—  0.0020 

88 

0.00318 

0.69118 

it 

It 

11678.11 

14.7880 

3.3333 

1 

i 

126  EXPERIMENTS   ON  THE   FLOW  OF   WATER  OVER  WEIRS. 


COMPARISON   OF  THE  PROPOSED  FORMULA  WITH   THE  RESULTS   OBTAINED   BY  PREVIOUS 

EXPERIMENTERS. 

160.  We   find   on   record   a   great   number    of    experiments    on    the    discharge 
of  water  over  weirs ;    in  the   present  state   of  the   science   of  hydraulics,  however, 
a  large   proportion   of  them   can   be   considered   only  in   the   light   of  first  approxi- 
mations;    of    great   value    undoubtedly,    at    the     respective     epochs    at   which   they 
were   made;    but   it   could   serve    no    useful    purpose    to    compare    them    with   the 
results  obtained  with   the   more   perfect   apparatus  used   of   late  years.      Three   sets 
of    experiments    have    been    made    in    France   within    the   last   thirty   years,    on   a 
comparatively  minute   scale,  it  must   be  admitted,  but  with  complete  apparatus,  and 
conducted   with   great   care.       They   were   made   by   Poncelet   and   Lesbros  at   Metz, 
in    1827  and    1828;    by  Castel    at   Toulouse,  in  1835;    and   by  Boileau   at  Metz,  in 
1846.     It  will  be   recollected   that   the   application  of  the   proposed  formula  to   the 
discharge    over  weirs   in  which   the   contraction   at   the    ends  is  complete,  is  limited 
to   depths   on  the  weir,  not  exceeding   one  third   of  the   length  of  the  sheet;    this 
limitation   permits   the  comparison   to   be   made  with    only  a   portion   of  the   results 
obtained    by  Poncelet   and   Lesbros,  and   by   Castel.      Boileau   operated   on   weirs  in 
which   the    end    contraction   was   suppressed,  and   to  which  form   the   limitation   does 
not   apply. 

161.  Comparison   of  the  proposed  formula,  with   the   results  obtained  by  Poncelet   and 
Lesbros.      These    experiments   are   to   be   found   among    the   magnificent   series   made 
at  the    expense    of  the   French   Government,  and   recorded   at   length   in   Experiences 
hydrauliques  sur  les  lois  de  I'ecoulement  de  I'eau  by  M.  M.  Poncelet  and  Lesbros,  Paris : 
1832;     and    in    the    continuation    under    the    same    title    by    M.    Lesbros,     Paris: 
1851.      In   table   XXXIX.,   of  the   last  mentioned   work,   are   given   the   coefficients 
for   computing    the    discharge    over  weirs   of    a    variety   of   forms,    and    of    certain 
lengths,  and  with  certain  depths  of  water,  by  the  formula 


in  which  d  is  the  discharge,  m  the  coefficient,  I  the  length,  h  the  depth,  and 
g=  9.8088  metres,  or  32.1817  feet.  The  comparison  can  be  usefully  made  with 
only  one  of  the  forms  experimented  upon,  namely :  that  in  which  the  orifice 
was  made  in  a  thin  plate,  in  the  plane  side  of  a  reservoir;  the  orifice  being  at 
a  great  distance  from  the  bottom  and  lateral  sides,  and  the-  discharge  made 
freely  into  the  air. 


EXPERIMENTS   ON  THE   FLOW   OF   WATER   OVER  WEIRS.  127 

In  table  XIV.  are  given  the  quantities  computed  according  to  Lesbros,  for 
all  the  depths  for  which  he  gives  values  of  m,  determined  by  experiment,  and 
which  are  within  the  limitation  required  by  the  proposed  formula,  namely :  that 
the  depth  shall  not  exceed  one  third  of  the  length.  The  quantities  are  also 
given  as  computed  by  the  proposed  formula.  It  will  be  perceived  by  the  final 
column  of  the  table,  that  the  proportional  differences  are  nearly  constant,  and 
that  the  quantities  by  the  proposed  formula  are  too  small  by  a  little  more  than 
two  per  cent.  If  the  coefficient  of  the  proposed  formula  was  changed  from  3.33 
to  3.41,  the  computed  results  would  agree  very  nearly.  It  should  be  recol- 
lected that  the  constants  in  the  proposed  formula  have  been  determined  from 
experiments  in  which  the  depths  upon  the  weir  were  from  0.6  to  1.6  feet,  or 
about  eight  times  the  depths  in  the  experiments  by  Poncelet  and  Lesbros.  It 
is  the  general  result  of  all  the  precise  experiments  on  the  discharge  through 
openings  of  a  variety  of  forms,  in  a  thin  plate,  that,  for  very  small  heads,  the 
coefficients  require  to  be  increased;  which  proves  that  the  law  of  the  discharge 
varying  as  the  square  root  of  the  head,  does  not  hold  good  for  very  small 
heads.  The  comparison  in  table  XIV.  affords  the  same  indications;  and  the 
constancy  of  the  proportional  differences,  indicates  that  the  correction  of  the 
length,  to  compensate  for  the  effect  of  the  end  contraction,  is  practically  correct, 
both  for  large  and  small  depths  upon  the  weir.  It  would  not  be  difficult  so 
to  determine  the  values  of  the  constants  in  the  formula 

Q=  C(l—lnh)htt, 

as  to  represent  the  experiments  both  of  Poncelet  and  Lesbros  and  the  Lower 
Locks  experiments  with  nearly  the  same  degree  of  exactness  that  the  latter  are 
represented,  with  the  constants  that  have  been  adopted.  This  would  undoubt- 
edly be  an  advantage  in  some  particular  cases  in  practice,  but  if  it  was 
intended  to  make  the  formula  general,  the  sacrifice  of  simplicity  would  be 
more  than  an  equivalent  disadvantage. 


128 


EXPERIMENTS  ON  THE  FLOW  OF  WATER  OVER  WEIRS. 


TABLE    XIV. 

The  length  of  the  weir  is  constant,  and  equal  to  0.6562  feet. 


1 

a 

3 

4 

6 

Quantity  of  water  dis- 

Value of  the 

charged  by  the  formula 

Quantity  of  water  dis- 

Proportional difference, 

Depth  on 

coefficient  m 

Q  =  m/A  v'2?  *> 

charged  by  the  formula 

or  the  absolute  dif- 

the weir. 

according  to 

m  haying  the  correspond- 

a. 

ference  divided  by  the 

Lesbros. 

ing  value  in  the  pre- 

Q =  3.33(  L—0.\nH)H  »  . 

quantity  in  column  8. 

ceding  column. 

Feet. 

Cubic  feet  per  second. 

Cubic  feet  per  second. 

0.06562 

0.417 

0.0369 

0.0360 

—  0.0245 

0.08202 

0.414 

0.0512 

0.0500 

—  0.0225 

0.09843 

0.412 

0.0670 

0.0655 

—  0.0228 

0.11483 

0.409 

0.0838 

0.0820 

—  0.0207 

0.13124 

0.407 

0.1019 

0.0997 

—  0.0209 

0.14764 

0.405 

0.1210 

0.1184 

—  0.0212 

0.16404 

0.404 

0.1413 

0.1379 

—  0.0239 

0.18045 

0.402 

0.1622 

0.1583 

—  0.0243 

0.19685 

0.401 

0.1844 

0.1794 

—  0.0271 

0.21326 

0.399 

0.2069 

0.2012 

—  0.0274 

162.  Comparison  of  the  proposed  formula  with  the  results  obtained  ly  Castel.  An 
abstract  of  these  experiments  may  be  found  in  the  Annaks  de  Ckimie  et  de 
Physique,  vol.  62.  Paris :  1836 ;  and  in  the  Annaks  des  ponts  et  chaussees,  vol.  1,  for 
1837.  Paris.  It  appears  to  have  been  a  leading  idea  in  these  experiments,  to 
imitate,  as  nearly  as  possible,  the  forms  and  proportions  of  the  weirs  ordinarily 
used  in  practice  for  gauging  streams  of  water;  in  fact,  to  reproduce  them  on  a 
small  scale,  anticipating  that  the  rules  deduced  from  precise  experiments  upon 
them  might  be  applied,  without  modification,  to  gaugings  on  a  large  scale.  The 
weir  was  formed  by  damming  up  a  wooden  canal,  2.4279  feet  in  width,  by  a 
thin  plate  of  copper,  in  which  the  weir  was  formed,  the  crest  being  0.5578  feet 
above  the  bottom  of  the  canal;  the  width  of  the  weir  varying  from  about  i  of 
a  foot  to  2i  feet.  The  latter  width  is  so  near  that  of  the  canal,  that  the  end 
contraction  must  have  been  sensibly  modified,  so  that  any  comparison  of  the 
results  obtained  from  it  would  be  of  little  use;  they  have  consequently  been 
omitted.  In  the  abstract  referred  to,  a  table  is  given  of  the  coefficients  deduced 
from  the  experiments,  for  a  variety  of  widths  and  depths.  In  table  XV.  are 
given  the  quantities  computed  with  these  coefficients,  for  all  the  widths  and 
depths  to  which  the  proposed  formula  is  applicable ;  also  the  quantities  as  com- 
puted by  the  proposed  formula.  In  consequence  of  the  small  dimensions  of  the 
canal,  the  water  approaching  the  weir  had  a  sensible  velocity;  in  table  XV. 


EXPERIMENTS   ON   THE   FLOW   OF   WATEK   OVER  WEIRS.  129 

the  depths  on  the  weir,  for  which  the  quantities  have  been  computed  by  the 
proposed  formula,  have  been  corrected  for  this  velocity.  It  will  be  seen  by 
referring  to  the  final  column,  that  the  proportional  differences  are  considerably 
greater,  and  have  less  uniformity  than  in  the  comparison  with  the  experiments 
of  Poncelet  and  Lesbros;  nevertheless,  there  is  a  certain  harmony  in  the  results 
of  both  comparisons,  and  they  serve  to  show  how  unsafe  it  is,  in  the  present 
state  of  the  science  of  hydraulics,  to  apply  rules  to  gauging  streams  of  water 
passing  over  weirs,  of  which  the  dimensions  differ  greatly  from  those  in  the 
experiments  from  which  the  rules  have  been  deduced. 

17 


130 


EXPERIMENTS   ON   THE   FLOW   OF   WATER  OVER  WEIRS. 


TABLE     XV. 

Width  of  the  canal  leading  to  the  weir  2.4279  feet ;    height  of  the  crest  of  the  weir  above  the  bottom  of 

the  canal  0.5578  feet. 


1 

2 

3 

4 

5 

6 

7 

8 

_ 

Quantity  of  water 

Head  due  the 

Depth  on  the  weir, 

Value  of  the  coeffi- 
cient m,  in  the 

discharged  by  the 
formula 

mean  velocity  of 
the  water  in  the 

corrected  for  the 

Telocity  of  the  water 

Quantity  of  water 
discharged  by  the 

Proportional  differ- 
ence, or  the  abso- 

Length of  the 

Depth  on  the 

formula 

r  • 

canal  leading  to 

in  the  canal  by  the 

formula 

lute  difference  of 

weir. 

weir. 

ITfpJn^Tf 

Q=m$LHy2gJ1, 

the  weir  by  the 

formula 

Q= 

the  quantities  in 

"       *       V*s     i 

m  having  the  corre- 

formula 

1V= 

3 

columns  4  and  7, 

according  to 
Castel 

sponding  value  in  the 

h    - 

11  W__LfcW'       7  V  I"^ 

B,23(L-Q.lnH')&? 

divided  by  the 
quantity  in  col- 

64.373 

1  \fl~^-fl)  JJ  —  n  £  j  A 

umn  4. 

Feet. 

Feet. 

Cubic  feet  per  second. 

Feet. 

Feet. 

Cubic  feet  per 

second. 

0.3281 

0.09843 

0.618 

0.0335 

0.00001 

0.09844 

0.0317    , 

—  0.0537 

0.6562 

0.19685 

0.604 

0.1852 

0.00016 

0.19701 

0.1796 

—  0.0302 

« 

0.16404 

0.611 

0.1425 

0.00010 

0.16414 

0.1380 

—  0.0311 

(t 

0.13124 

0.619 

0.1033 

0.00006 

0.13130 

0.0998 

—  0.0339 

u 

0.09843 

0.624 

0.0676 

0.00003 

0.09846 

0.0655 

—  0.0318 

0.9843 

0.32809 

0.604 

0.5976 

0.00120 

0.32924 

0.5778 

—  0.0331 

u 

0.26247 

0.606 

0.4290 

0.00072 

0.26316 

0.4189 

—  0.0237 

u 

0.19685 

0.610 

0.2805 

0.00036 

0.19720 

0.2755 

—  0.0176 

u 

0.16404 

0.616 

-      0.2155 

0.00023 

0.16426 

0.2109 

—  0.0211 

u 

0.13124 

•'0.623 

0.1559 

0.00014 

0.13138 

0.1519 

—  0.0257 

u 

0.09843 

0.631 

0.1026 

0.00006 

0.09849 

0.0993 

—  0.0322 

1.3124 

0.39371 

0.621 

1.0769 

0.00337 

0.39687 

1.0266 

—  0.0468 

(I 

0.32809 

0.621 

0.8192 

0.00225 

0.33022 

0.7876 

—  0.0386 

« 

0.26247 

0.620 

0.5852 

0.00134 

0.26375 

0.5682 

—  0.0291 

ft 

0.19685 

0.622 

0.3813 

0.00067 

0.19749 

0.3720 

—  0.0245 

a 

0.16404 

0.626 

0.2920 

0.00043 

0.16446 

0.2842 

—  0.0266 

u 

0.13124 

0.632 

0.2109 

0.00025 

0.13148 

0.2042 

—  0.0320 

a 

0.09843 

0.636 

0.1379 

0.00012 

0.09855 

0.1332 

—  0.0341 

1.6404 

0.32809 

0.631 

1.0405 

0.00363 

0.33147 

1.0003 

—  0.0386 

u 

0.26247 

0.632 

0.7457 

0.00218 

0.26452 

0.7192 

—  0.0355 

" 

0.19685 

0.632 

0.4843 

0.00108 

0.19788 

0.4692 

—  0.0312 

(t 

0.16404 

0.633 

0.3690 

0.00069 

0.16470 

0.3578 

—  0.0304 

u 

0.13124 

0,636 

0.2653 

0.00039 

0.13161 

0.2566 

—  0.0327 

u 

0.09843 

0.642 

0.1740 

0.00019 

0.09861 

0.1671 

—  0.0393 

1.9685 

0.32809 

0.644 

1.2743 

0.00545 

0.33308 

1.2174 

—  0.0446 

u 

0.26247 

0.644 

0.9118 

0.00326 

0.26549 

0.8725 

—  0.0431 

" 

0.19685 

0.645 

0.5931 

0.00163 

0.19838 

0.5675 

—  0.0432 

« 

0.16404 

0.644 

0.4505 

0.00103 

0.16502 

0.4320 

—  0.0410 

ii 

0.13124 

0.645 

0.3229 

0.00058 

0.13179 

0.3094 

—  0.0417 

H 

0.09843 

0.651 

0.2117 

0.00027 

0.09869 

0.2012 

—  0.0495 

EXPERIMENTS   ON   THE   FLOW   OF   WATER  OVER  WEIRS.  131 

163.  Comparison  of  the  proposed  formula,  imth  that  obtained  by  Boileau.  The 
experiments  from  which  Boileau  deduced  his  formula,  are  given  at  length  in 
Jaugcage  des  cours  d'eau  a  foible  ou  a  moyenne  section,  by  M.  P.  Boileau.  Paris :  1850. 
Boileau  has  particularly  studied  the  discharge  in  the  form  of  weir  in  which  the 
contraction  at  the  ends  is  suppressed ;  that  is  to  say,  the  form  in  which  the 
weir  occupies  the  whole  width  of  the  canal  conducting  the  water  to  it.  The 
proposed  formula  is  applicable  to  this  case,  by  making  n  =  0.  Boileau  experi- 
mented on  three  weirs  of  this  form ;  one  of  them  was  5.30  feet  in  length,  with 
the  crest  1.54  feet  above  the  bottom  of  the  canal ;  the  other  two  were  2.94 
feet  in  length,  the  crest  in  one  being  1.12  feet  above  the  bottom  of  the  canal ; 
and  in  the  other  1.61  feet  above  the  bottom ;  the  depths  on  the  weir  varying 
from  0.19  feet  to  0.72  feet.  By  a  train  of  reasoning  combined  with  the  results 
of  his  experiments,  Boileau  has  arrived  at  the  following  formula  for  weirs  of  this 
form :  — 


in   which 


Q  =  the    discharge. 

$i=the   height  of    the    crest    of    the   weir,   above    the   bottom   of    the   canal, 

which  is   supposed   to  be   horizontal  for   a   short  distance,  upstream  from 

the  weir. 
JET=  the   depth   on   the   weir,   taken  before   the   sheet  begins   to   curve   in   con- 

sequence  of  the    discharge. 

Z:=the  width  of  the  canal,  and  also  the  length  of  the  weir. 
g  =  9.8088"1. 

The  coefficient  0.417  is  determined  from  a  mean  of  14  experiments. 
Adopting  the  English  foot  as  the  unit,  and  reducing,  we  have 


For  this  form  of  weir,  the  proposed  formula  becomes 


:  (B) 

H'  being    the    depth    upon    the    weir,    corrected    for    the  velocity   of    the    water 
approaching  the   weir. 

These   formulas   differ   so   essentially   that   they   can    be  conveniently   compared 


132  EXPERIMENTS   ON  THE    FLOW   OF   WATER  OVER   WEIRS. 

only  by  applying  them   to   particular    cases.      In   the   formula   (.4),   as   &  increases 

O      I         TT 

relatively   to   H,   the    factor   ,--j--====  approaches  unity,  which  is  the  limit  when 


8  is  infinitely  greater  than  H;  in  the  '  latter  case  we  have  also,  H'  =:  H;  the 
formulas  (A)  and  (B)  then  become  identical,  excepting  the  coefficients,  that  in  (B) 
being  vfa  less  than  in  (A).  Hence  we  may  conclude  that  for  any  length  of 
weir,  and  for  any  depth  upon  it,  providing  that  the  depth  of  the  canal  leading 
to  the  weir,  is  very  great  relatively  to  the  depth  on  the  weir,  the  quantities 
computed  by  the  formulas  (A)  and  (B)  will  differ  ^|^  only. 

In  practice,  however,  S  is  seldom  very  great,  relative  to  H.  Let  us  take  an 
example  conforming  more  nearly  to  the  usual  cases  that  occur  in  practice.  Let 
ff=l  foot,  #=3  feet,  L  =  10  feet,  by  the  formula  (A),  Q=  34.552  cubic  feet 
per  second.  In  the  formula  (B\  H'  is  the  depth  on  the  weir,  corrected  for  the 
mean  velocity  of  the  water  approaching  the  weir  ;  this  velocity  is  equal  to  the 
quotient  of  the  area  of  the  section  of  the  canal,  divided  by  the  quantity.  But 
the  quantity  itself  depends  on  this  velocity.  The  formula  (.Z?),  if  put  under  a 
form  to  give  the  quantity  directly  from  the  measured  depth  upon  the  weir, 
would  become  very  complicated  ;  it  will  be  equally  exact  and  much  easier,  to 
find  the  quantity  by  successive  approximations  as  follows. 

1st  approximation. 

Assume  H'  =  1,  then   Q  =  33.3. 
2nd  approximation. 

If   Q  =  33.3,  the   mean   velocity   of    the   water    in   the    canal    leading    to    the 

oo  o 

weir  is     .  „'.  „.  =  0.8325  ;    and  for  the   head   due   this  velocity  we   have 


— /       =1.0103; 


Q  =  33.810. 

A   third   approximation  in   a   similar   manner   gives    Q  =  33.817. 

The  proportional  difference  of  the  quantities  by  the  two  formulas  is  about 
•£r,  or  a  little  over  two  per  cent. 

Boileau,  in  establishing  his  formula,  assumes  that  the  living  force  in  the 
entire  section  of  the  canal  is  expended  in  increasing  the  discharge  over  the 


EXPERIMENTS   ON  THE   FLOW   OF   WATER   OVER   WEIRS.  133 

weir;  in  the  method  adopted  in  this  work  for  correcting  the  depth  on  the  weir 
for  the  velocity  of  the  water  in  the  canal,  it  is  assumed  that  the  living  force 
in  the  part  of  the  section  of  the  canal  equal  to  the  area  of  the  orifice  of  dis- 
charge only,  is  expended  in  increasing  the  discharge ;  as  applied  to  a  weir  of 
the  form  under  consideration,  it  is  clear  that  neither  of  these  assumptions  is 
strictly  true ;  the  latter,  however,  appears  to  be  the  most  rational,  and  to  agree 
the  best  with  experiment. 


PRECAUTIONS   TO  BE   OBSERVED   IN  THE   APPLICATION  OF  THE   PROPOSED  FORMULA. 


164.  Q=:3.33(L  — 

in  which 

Q  =  the  discharge,  in  cubic  feet  per  second  ; 
L  =  the  length  of  the  weir  ; 
n  =  the  number  of  end  contractions  ; 
H—  the  depth  on  the  weir  ; 

the  English  foot  being  the  unit  of  measure. 

When  the  contraction  is  complete  at  each  end  of  the  weir,  n  =  2  ;  when  the 
weir  is  of  the  same  width  as  the  canal  conducting  water  to  it,  the  end  con- 
traction is  suppressed,  and  n  =  0. 

This  formula  is  only  applicable  to  rectangular  weirs,  made  in  the  side  of  a 
dam,  which  is  vertical  on  the  upstream  side,  the  crest  of  the  weir  being  hori- 
zontal, and  the  ends  vertical  ;  also,  the  edges  of  the  orifice  presented  to  the  cur- 
rent must  be  sharp;  for,  if  bevelled  or  rounded  off  in  any  perceptible  degree,  a 
material  effect  will  be  produced  on  the  discharge  ;  it  is  essential,  moreover,  that 
the  stream  should  touch  the  orifice  only  at  these  edges,  after  passing  which  it 
should  be  discharged  through  the  air,  in  the  same  manner  as  if  the  orifice  was 
cut  in  a  thin  plate.  See  fig.  3,  plate  XVIII. 

The  formula  is  not  applicable  to  cases  in  which  the  depth  on  the  weir 
exceeds  one  third  of  the  length  ;  nor  to  very  small  depths.  In  the  experiments 
from  which  it  has  been  determined,  the  depths  have  varied  from  7  inches  to 
nearly  19  inches,  and  there  seems  no  reason  why  it  should  not  be  applied  with 
safety  to  any  depths  between  6  inches  and  24  inches. 

The   height    of    the    surface    of   the   water    in    the   canal,   above   the   crest   of 


134  EXPERIMENTS   ON  THE   FLOW  OF   WATER  OVER   WEIRS. 

the  weir,  is  to  be  taken  for  the  depth  upon  the  weir;  this  height  should  be 
taken  at  a  point  far  enough  from  the  weir  to  be  unaffected  by  the  curvature 
caused  by  the  discharge ;  if  more  convenient,  it  may  be  taken  by  means  of  a 
pipe  opening  near  the  bottom  of  the  canal  near  the  upstream  side  of  the  weir, 
which  pipe  may  be  made  to  communicate  with  a  box  placed  in  any  convenient 
situation ;  and  if  the  box  and  pipe  do  not  leak,  the  height  may  be  observed,  in 
this  manner,  very  correctly  (art.  175).  However  the  depth  may  be  observed,  it 
may  require  to  be  corrected  for  the  velocity  of  the  water  approaching  the  weir. 

The  end  contraction  must  either  be  complete,  or  entirely  suppressed;  the 
necessary  distance  from  the  side  of  the  canal  or  reservoir  to  the  end  of  the 
weir,  in  order  that  the  end  contraction  may  be  complete,  is  not  definitely  deter- 
mined; in  experiments  1  to  4,  table  XIII.,  the  depth  on  the  weir  was  about 
1.5  feet,  and  the  distance  from  the  side  of  the  canal  to  the  end  of  the  weir, 
about  2  feet;  the  proposed  formula  applies  well  to  all  these  experiments.  In 
cases  where  there  is  end  contraction,  we  may  assume  a  distance  from  the  side 
of  the  canal  to  the  end  of  the  weir  equal  to  the  depth  on  the  weir,  as  the 
least  admissible,  in  order  that  the  proposed  formula  may  apply. 

As  to  the  fall  below  the  weir,  requisite  to  give  a  free  discharge  to  the 
water,  it  is  not  definitely  determined ;  a  comparison  of  experiments  49,  50,  and 
51,  table  X.,  indicates  that,  when  the  depth  on  the  weir  is  1  foot,  and  the  entire 
sheet,  after  passing  the  weir,  strikes  a  solid  body  at  about  0.5  feet  below  the 
crest  of  the  weir,  the  discharge,  with  the  same  depth,  is  diminished  about  j^nr- 
By  experiments  1  and  2,  table  XII.,  it  appears  that,  when  the  sheet  passing  the 
weir,  falls  into  water  of  considerable  depth,  the  depth  on  the  weir  being  about 
0.85  feet,  no  difference  is  perceptible  in  the  discharge,  whether  the  water  is  1.05 
feet  or  0.235  feet  below  the  crest  of  the  weir ;  it  is  very  essential,  however,  in 
all  cases,  that  the  air  under  the  sheet  should  have  free  communication  with  the 
external  atmosphere.  With  this  precaution  it  appears  that,  if  the  fall  below  the 
crest  of  the  weir  is  not  less  than  half  the  depth  upon  the  weir,  the  discharge 
over  the  weir  will  not  be  perceptibly  obstructed.  If  the  sheet  is  of  very  great 
length,  however,  more  fall  will  be  necessary,  unless  some  special  arrangement  is 
made  to  supply  air  to  the  space  under  the  sheet  at  the  places  that  would 
otherwise  not  have  a  free  communication  with  the  atmosphere. 

In  respect  to  the  depth  of  the  canal  leading  to  the  weir,  experiments  36  to 
43,  table  XIII.,  show  that,  with  a  depth  as  small  as  three  times  that  on  the 
weir,  the  proposed  formula  agrees  with  experiment,  within  less  than  one  per  cent. ; 
this  proportion  may  be  taken  as  the  least  admissible,  when  an  accurate  gauging 
is  required. 


EXPERIMENTS   ON  THE   FLOW   OF   WATER  OVER   WEIRS.  135 

It  not  unfrequently  happens  that,  in  consequence  of  the  particular  form  of 
the  canal  leading  to  the  weir,  or  from  other  causes,  the  velocity  of  the  water 
in  the  canal  is  not  uniform  in  all  parts  of  the  section;  this  is  a  frequent  cause 
of  serious  error,  and  is  often  entirely  overlooked.  If  great  irregularities  exist, 
they  should  be  removed  by  causing  the  water  to  pass  through  one  or  more 
gratings,  presenting  numerous  small  apertures  equally  distributed,  or  otherwise,  as 
the  case  may  require,  through  which  the  water  may  pass  under  a  small  head ; 
these  gratings  should  be  placed  as  far  from  the  weir  as  practicable. 

If  the  canal  leading  to  the  weir  has  a  suitable  depth,  it  will  be  requisite 
only  when  great  precision  is  required,  to  correct  the  depth  upon  the  weir  for 
the  velocity  of  the  water  in  the  canal  by  the  formula  (Z>)  (art.  153) ;  thus,  in 
experiment  42,  table  XIII.,  the  water  in  the  canal  had  a  mean  velocity  of  about 
1  foot  per  second,  the  effect  of  which  was  to  increase  the  discharge  about  two 
per  cent. ;  in  experiment  82,  in  which  the  velocity  was  about  0.5  feet  per 
second,  the  discharge  was  increased  about  one  per  cent. ;  these  examples  will 
enable  the  operator  to  judge,  in  each  case,  of  the  necessity  of  going  through  the 
troublesome  calculation  for  correcting  the  depth  on  the  weir. 


MISCELLANEOUS    EXPERIMENTS    ON    THE    FLOW   OF    WATER,   MADE   AT 
THE  LOWER  LOCKS,  IN  NOVEMBER,  1852. 


On  the  discharge  of  water  over  a  dam  of  the  same  section  as  that   erected  by  the  Essex  Company,  across  the 

Merrimack  River  at  Lawrence,  Massachusetts. 

165.  As   these   experiments  cannot   be   usefully  compared  with   those  on  weirs 
of  more   regular  form,  they  have   not   been   included   in  table  XIII. ;    and   as  they 
are  of  less  general  interest,  they  will  not  be  given  with  much  detail. 

The  form  of  the  dam  is  represented  by  figures  11  and  12,  plate  XIV. 
(art.  147) ;  the  other  apparatus  was  the  same  as  that  used  for  the  experiments 
in  table  XIII. 

The  end  contraction  was  suppressed  by  making  the  canal  leading  to  the 
overfall  of  the  same  width  as  the  overfall  itself.  The  water  in  the  hook  gauge 
boxes  communicated  only  with  the  water  contained  in  the  spaces  between  the 
masonry  and  the  wood-work  forming  the  sides  and  bottom  of  the  canal  leading 
to  the  overfall;  as  there  was  a  free  communication  between  the  water  at  A, 
figures  11  and  12,  and  that  near  the  hook  gauge  boxes,  and  as  the  water 
between  these  places  was  sensibly  at  rest,  we  may  consider  that  the  height  of 
the  water  was  taken  at  A. 

166.  In  table  XVI.  these  experiments  are  exhibited   in   sufficient  detail   to   be 
intelligible. 

COLUMNS  1  and  2  require  no  explanation. 

COLUMN  3.  The  heights  contained  in  this  column  are  above  the  mean  level 
of  the  crest  of  the  dam,  which  was  very  nearly  horizontal  for  a  distance  of 
2.95  feet  from  C  to  D.  These  heights  have  not  been  corrected  for  the  velocity 
of  the  water  approaching  the  weir;  indeed,  from  the  manner  in  which  they 
were  observed,  no  correction  was  necessary. 

,    COLUMN  4.     The  quantities  in   this   column   have   been   obtained  in  the   manner 
described   in   the   explanation   of  table   XIII.  (art.  155). 

COLUMN  5.     Quantity  of  water  passing  over  the  dam,  calculated  by  the  formula 


.53 


EXPERIMENTS  ON   THE   FLOW  OF   WATER  OVER  WEIRS. 


137 


This  formula  was  arrived  at  by  trial  of  various  powers  of  h,  and  was 
adopted  as  representing,  the  most  nearly,  the  results  of  the  five  experiments  in 
the  table ;  it  should  be  distinctly  understood,  however,  that  it  is  not  applicable 
to  depths  much  greater  or  less  than  in  the  experiments  from  which  it  is 
deduced.  In  April,  1852,  the  depth  of  water  flowing  over  the  dam  at  Law- 
rence, was  10  feet ;  if  the  quantity  then  passing  over  the  dam  was  com- 
puted by  this  formula,  it  is  probable  that  it  would  be  greatly  in  error. 

x  COLUMN  6.  Proportional  difference.  It  will  be  observed  that  the  greatest  pro- 
portional difference  is  0.0085,  or  less  than  one  per  cent. ;  we  may  therefore  say 
with  confidence,  that  we  can  compute  the  flow  of  water  over  the  Lawrence  dam, 
when  free  from  ice  or  other  obstruction,  for  any  depth  not  greater  than  20 
inches  or  less  than  7  inches,  without  being  liable  to  an  error  exceeding  one 
per  cent. 


TABLE    XVI. 


Time,  from  November  10th,  8A,  57',  P.M.,  to  November  llth,  OJ.  11',  A.  H. 

Temperature  of  the  air  at  10»,  5W,  P.  M.,  34.50°  Fahrenheit. 

"               "      water  "      "      "       41.75"        " 

The  air  culm. 

1 

3 

3 

4 

5 

6 

Number  of 
the  experi- 
ment 

Length  of  the 

ova-full. 

I. 

Mean  height  of  the 
surface  of  the  water 
in  the  hook  gauge 
boxes,  above  the  top 
of  the  horizontal 
crest  of  the  dam. 

Feet. 

Quantity  of  water 
passing  over  the 
dam,  as  measured 
in  the  lock  cham- 
ber.   In  cubic  feet 
per  second. 

Quantity  of  water 
passing  over  the 
dam  calculated  by 
the  formula 

Q  =  3.01208  lliM 
In  cubic  feet  per 

Proportional  difference, 
or  the  absolute  differ- 
ence of  the  quantities 
in  columns  4  and  5, 
divided  by  the  quantity 
in  column  4. 

h. 

second. 

89 

9.995 

0.58720 

13.385 

13.332 

—  0.0040 

90 

a 

0.79035 

20.892 

21.005 

4-0.0054 

91 

a 

0.97670 

28.914 

29.039 

4-0.0043 

92 

a 

1.32520 

46.183 

46.317 

4-0.0029 

93 

u 

1.63380 

64.346 

63.804 

—  0.0085 

EXPERIMENTS  TO  ASCERTAIN  THE  EFFECT  OF  TAKING  THE  DEPTHS  UPON  A  WEIR  AT  DIFFERENT 
DISTANCES  FROM  IT,  BY  MEANS  OF  PIPES  OPENING  NEAR  THE  BOTTOM  OF  THE  CANAL. 

167.  It  is  often  a  matter  of  great  doubt  and  uncertainty,  to  know  at  what 
distance  from  the  weir  the  depth  of  the  water  upon  it  should  be  observed; 
very  often  also  it  becomes  a  matter  of  necessity  to  observe  the  depth  at  a  dis- 
tance from  the  weir  so  small  that,  according  to  some,  the  quantity  of  water 
passing  the  weir,  computed  in  the  usual  manner,  would  be  liable  to  sensible 

18 


138  EXPERIMENTS   ON   THE   FLOW   OF   WATER  OVER  WEIRS. 

error.  For  the  purpose  of  obtaining  some  light  upon  this  point  these  experi- 
ments were  undertaken,  and,  as  they  were  made  with  all  the  precautions  for 
insuring  accuracy  that  could  be  devised,  they  will  be  described  with  some  detail. 

168.  Figures  8,  9,  and  10,  plate  XIV.,  represent  the  form  of  the  weir,  and  the 
system   of  pipes   used   for   these    experiments.      The    canal   leading   to  the  weir  was 
of    the    same   width    as    the    weir,   so    that    the    end    contraction    was    suppressed. 
The   pipes  were    of    lead,   about   three    fourths    of    an    inch   interior   diameter,   the 
lower   extremities    of   which,    numbered    from    1    to     8,    were    about    three    inches 
above   the    bottom    of    the    canal,    and   terminated   in   holes   in   the   board    CO;    the 
side   of   the    board    at   which    they   opened    was    vertical,    and    in   the    axis    of    the 
canal;    the  ends  of  the  pipe  did  not  project  through  the  board;    the  other  extremi- 
ties   of    the    pipes   were    fastened    by   small    flanges   to   the   bottoms   of    the    hook 
gauge    boxes ;     holes  were    made   in    the    bottoms   of    the    boxes   corresponding  to 
each   pipe,    and    communication    between   the    boxes   and   the    pipes   could     be    con- 
trolled   at    pleasure,    by   plugging    up    these    holes.      It   will    be    readily   perceived 
that    heights    of   the   water    observed    by   this    apparatus    are    not    necessarily   the 
true   elevations   of   the   surface   of    the   water   immediately   over   the   orifices   of  the 
pipes,  but   that   they  are   the    elevations   of  the    surface   in   the    hook   gauge  boxes; 
an  elevation  which  is  due  to  the  statical  pressure  on  the  orifice  of  the  pipe. 

169.  In   order   to    obtain    the    heights    at    different    distances    from    the   weir, 
observations   were    necessarily   made   with    both    hook    gauges    at    the    same    time, 
one   of  which   was   always   in    communication  with   a   pipe    opening   at    6    feet  from 
the   weir,    the    apertures   in   the    bottom    of    the   box,    communicating   with   all    the 
other   pipes,    being   plugged    up ;     at   the   other   hook   gauge,    either   pipe   might   be 
in   communication    with   the   box,  all   the    other   apertures   being   plugged   up ;    thus, 
the   depth   at    six    feet    from    the   weir    was    observed    in    each    experiment,    to   be 
used   as   a   standard   with   which   the    depth    observed    simultaneously   at   any    other 
distance   might   be   compared ;    this   mode   of  proceeding  was   rendered   necessary,  in 
consequence    of    the    impossibility   of    maintaining   the    level   of    the   water   uniform 
for   any   considerable   length    of  time. 

170.  In   considering   the  sources   of  error    to  which   the   observations  with   the 
hook   gauges   were   liable,    it    appeared    that    four    kinds    required    to    be    specially 
guarded    against,   namely :    First,  imperfect   comparison    of    the    gauges,  with  the    top 
of  the   weir.      Second,  defective    stability,  in   consequence   of  which   the   relative    ele- 
vations  of  the   gauges   and   the  weir   might   not   be   constant.      Third,  errors   in  the 
graduation   of    the    gauges.      Fourth,   the    difference    in    the   habit    of    observers,   in 
making   the    point   of  the   hook   coincide   with    the    surface   of  the   water ;    or,  what 
we  may   call,  the   personal   error.      In   relation   to   the  first,  we  must  bear  in  mind 


EXPERIMENTS   ON   THE   FLOW   OF   WATER  OVER  WEIRS.  139 

that  the  requirement  here  is  not  so  much  that  the  absolute  height  above  the 
top  of  the  weir  should  be  exactly  determined,  as  that  the  difference  of  the 
heights  at  two  points,  at  different  distances  from  the  weir,  should  be  determined 
correctly ;  if  then  we  know  the  relative  heights  of  the  two  gauges,  the  object 
can  be  attained,  even  if  we  do  not  know  precisely  the  height  of  either  of  them, 
relatively  to  the  weir.  The  heights  of  the  gauges  relative  to  each  other,  could 
easily  be  ascertained  at  any  time,  by  closing  up  all  the  apertures  in  each  box, 
except  those  communicating  with  pipes,  numbers  4  and  5,  which,  it  will  be  seen 
by  reference  to  figure  9,  had  a  common  orifice  at  their  lower  extremities ;  con- 
sequently, the  surface  of  the  water  in  both  boxes  must  have  been  at  the  same 
level.  The  correction  to  be  applied  to  the  reading  of  one  of  the  hook  gauges, 
was  taken  as  previously  determined  for  the  experiments  on  the  discharge  over 
the  weir,  and  the  correction  for  the  other  gauge,  was  deduced  from  simultaneous 
observations  on  both  gauges,  when  the  boxes  communicated  with  a  common  ori- 
fice, in-  the  manner  just  described.  The  second  source  of  error  was  guarded 
against  as  much  as  practicable,  by  making  the  observations  for  the  correction 
just  described,  at  nearly  the  same  time  as  the  experiments  to  which  it  was  to 
be  applied.  The  danger  of  error  from  the  tUrd  source  was  much  diminished  by 
making  the  observations  for  the  correction,  with  nearly  the  same  depth  upon  the 
weir  as  in  the  experiments  to  which  it  was  to  be  applied.  The  fourth  source  of 
error  was  eliminated  by  determining  the  correction  separately  for  each  pair  of 
observers.  In  short,  these  four  sources  of  error  were  reduced  to  a  minimum  by 
determining  for  each  session  of  the  experiments,  and  for  each  pair  of  observers, 
the  relative  corrections  to  be  applied  to  the  readings  of  the  hook  gauges,  to 
give  the  depths  upon  the  weir;  the  depths,  when  the  observations  for  these 
corrections  were  made,  being  nearly  the  same  as  in  the  experiments  to  which 
they  were  to  be  applied. 

171.  In  table  XVII.  are  given  the  results  of  the  observations  made  for  the 
purpose  of  obtaining  the  relative  corrections  for  the  gauges,  for  each  session  of 
the  experiments,  and  for  each  pair  of  observers.  In  computing  the  depth  upon 
the  weir  by  the  north  hook  gauge,  the  correction  — 0.03072  is  applied  to  the 
mean  reading  of  the  gauge,  (art.  143) ;  the  mean  reading  of  the  south  hook 
gauge  is  given ;  as  the  water  in  both  boxes  is  at  the  same  height,  the  differ- 
ence between  the  depth  upon  the  weir,  as  determined  by  the  north  hook  gauge, 
and  the  mean  reading  of  the  south  hook  gauge,  must  give  the  correction  for  the 
last  named  gauge. 


140 


EXPERIMENTS   ON   THE   FLOW   OF   WATER  OVER   WEIRS. 


TABLE     XVII. 


DATE, 
1862. 

Time  of  beginning 
the  observation. 

North  hook  gauge,  in  communi- 
cation with  pipe  No.  5  opening 
near  the  bottom  of  the  canal  at 
6  feet  from  the  weir. 

South  hook  gauge,  in  communication  with  pipe  No.  4,  opening  near  the 
bottom  of  the  canal  at  6  feet  from  the  weir. 

Obserrer. 

Arithmetical 
mean  depth  on 
the  weir. 

Feet. 

Observer. 

Arithmetical 
mean  reading 
of  the  gauge. 

Feet. 

Correction  to  be 
applied  to  the  mean 
reading  to  give  the 
depth  on  the  weir. 

Feet. 

Mean  correction  for 
each  cession,  and 
each  pair  of 
observers. 

Feet. 

November  3. 

tt            tt 

9»  13'  P.M. 

10      0     " 

Francis 

K 

1.01180 
1.02617 

Avery 

a 

1.03760 

1.05375 

—  0.02580 

—  0.02758 

—  0.02669 

November  3. 

11*    16'   P.M. 

Haeffely 

1.00739 

Newell 

1.03377 

—  0.02638 

—  0.02638 

November  3. 

tt            tt 

«         4. 

9*  29'  P.M. 

10    45      « 

1      47     A.M. 

Francis 
" 
u 

1.01984 
1.01073 
1.04532 

Newell 

a 
tt 

1.04625 
1.03716 
1.07169 

—  0.02641 
—  0.02643 
—  0.02637 

—  0.02640 

November  3. 
«          4. 

11"      0'  P.M. 
1     58    A.M. 

Francis 

ft 

1.00807 
1.04734 

Haeffely 

tt 

1.03431 
1.07350 

—  0.02624 
—  0.02616 

—  0.02620 

November  7. 

u             a 
(t            a 
u            u 
u             u 
u            u 
tt             u 

M                  t( 

7"  50'  A.M. 
9    38     " 
2      7   P.M. 
8    22     " 
8    56     " 
9    28     " 
10      0     « 
10    31      « 

Francis 

u 

tt 
it 
U 
11 
it 

a 

0.98775 
0.98555 
0.73665 
1.00696 
1.00677 
1.00580 
0.99338 
0.99294 

Avery 

a 

u 
tt 
tt 
u 
u 
tt 

1.01362 
1.01195 
0.76357 
1.03287 
1.03311 
1.03244 
1.01973 
1.01961 

—  0.02587 
—  0.02640 
—  0.02692 
—  0.02591 
—  0.02634 
—  0.02664 
—  0.02635 
—  0.02667 

—  0.02639 

November  7. 

u             u 
u             u 

8»      4'  A.M. 

9    49     « 
2    26    P.M. 

Francis 

a 

a 

0.98932 
0.98019 
0.78315 

Newell 
0 

tt 

1.01478 
1.00597 
0.80906 

—  0.02546 
—  0.02578 
—  0.02591 

—  0.02572 

November  7. 

9*  20'  A.M. 

Haeffely 

0.99305 

Newell 

1.01997 

—  0.02692 

—  0.02692 

172.  It  will  be  perceived,  by  an  examination  of  table  XVII.,  that  there  are 
greater  irregularities  in  the  comparisons  by  some  observers,  than  in  those  by 
others;  this  is  to  be  attributed,  principally,  to  the  different  degrees  of  experi- 
ence and  skill  in  the  observers. 


EXPERIMENTS   ON   THE    FLOW   OF    WATER   OVER  WEIRS.  141 

173.  In   table    XVIII.    are    given   the    details   of  the    experiments,   to   ascertain 
the    effect   of  observing   the    depths   upon   the    weir,  at   different   distances   from   the 
weir,   by   means    of    pipes   opening    near    the    bottom   of    the    canal.      In   order   to 
obtain    the     depth    upon    the    weir     by     the     north     hook     gauge,    the    correction 
—  0.03072    has   been    applied   to  the   mean   readings   of  this  gauge.      The  correction 
for   the   south   hook    gauge    is    taken    from   the   final   column   of    table    XVII.,    for 
the   corresponding   session   and   pair   of  observers.     From  want  of  time,  pipes  num- 
ber 6  and  7  were  not  made  use  of. 

It  will  be  perceived,  by  referring  to  the  final  column  of  table  XVIII.,  that 
the  differences  in  the  heights,  at  the  different  distances  tried,  are  very  inconsid- 
erable, and  such  as  could  be  detected  only  by  the  most  delicate  means  of 
observation. 

174.  Two  comparisons  were   made   in   a  similar  manner,  of  the   heights,  when 
one    gauge    box    communicated    with    a    pipe    opening    near    the    bottom    of    the 
canal,    and    the    other   with    a    pipe    opening   through   the    side,    at   about    4.2    feet 
above   the   bottom,   the   orifices  of    both  being   at   6   feet  from   the   weir,  as   repre- 
sented  at   B,  figures  8,  9,  and  10,  plate   XIV. ;    the  following  are  the  results. 

First  comparison,  made  November  7th,  beginning  at  3h,  52',  P.M. 

Francis,  at  north  hook  gauge,  with  pipe  No.  5,  depth  on  weir         0.81616  feet. 

Avery,  at   south   hook   gauge,  with  pipe  B  0.81641     " 

Difference -f  0.00025  feet, 

Second  comparison,  made  November  7th,  beginning  at  4h,  5',  P.M. 

Francis,  at  north  hook  gauge,  with  pipe  No.  5,  depth  on  weir         0.81775  feet. 

Newell,  at  south  hook  gauge,  with  pipe  B  "  "  0.81776     " 

Difference +0.00001  feet. 

These  differences  are  so  minute  that  we  may  conclude  that  the  depth  was 
the  same  whether  the  pipe  opened  near  the  bottom  of  the  canal  or  at  4.2  feet 
above. 

175.  These    experiments,   taken    in    connection    with    those    of    Boileau,*   who 
has    arrived    at    similar    results,   leave    no    doubt    as   to    the    propriety,   whenever 
convenience    requires    it,   of    observing  the   depths   upon   the   weir  by  means   of    a 
pipe   opening   into  the  dead  water,  near  the   bottom  of  the  canal   on  the  upstream 
side   of  the   weir. 


'  Jaugeage  des  cours  d'eau,  by  M.  P.  Boikau.     Paris :  1850. 


142 


EXPERIMENTS  ON  THE   FLOW   OF   WATER  OVER   WEIRS. 


TABLE     XVIII. 


DATE, 

1852. 

Time  of 
beginning  the 
observation. 

North  hook  gauge. 

South  hook  gauge. 

Difference  in  the 
depths  upon  the 
weir,  the  pipe 
opening  at  6  feet 
from  the  weir 
being  the  standard. 

Mean  difference  in 
the  depths  upon 
the  weir,  the  pipe 
opening  at  6  feet 
from  the  weir 
being  the  standard. 

Pipe  No.5opensat  6fec 
"    "6       "8   « 
"    "    7       "     10   " 
a    .1    g        «     12    " 

t  from  the  weir. 

t<          tt 
a            u 

Pipe  No.l  opens  at  1  inch  from  the  weir. 
"    "    2        "     2  feet         "        " 
u    u    3       "4    "           "        " 

Num- 
ber of 
the 
pipe. 

Observer. 

Corrected 
depth  upon 
the  weir. 

Num- 
ber of 
the 
pipe. 

Observer. 

Corrected 
depth  upon 
the  weir. 

November  3 
«         4 
«         7 

u           u 
f<           « 

11*  53'  P.M. 
0  27  A.M. 

10  33     " 
10  44     " 
10  56     " 

I 

5 
5 

5 

Francis 
u 

Haeffely 

Francis 
tf 

1.01267 
1.01439 
0.97530 
0.97644 
0.97658 

1 
1 
1 
1 
1 

Newell 
Haeffely 
Newell 
Avery 
Newell 

1.01321 
1.01459 

0.97547 
0.97683 
0.97695 

+  0.00054 
+  0.00020 
--  0.00017 
--  0.00039 
-  -  0.00037 

+  0.00033 

November  4 

U                U 
U                « 

0*  37'  A.  M. 
1   23     " 
1   34     " 

5 
5 
5 

Haeffely 
Francis 
Haeffely 

1.02189 
1.04220 
1.04472 

2 
2 
2 

Newell 
Haeffely 
Newell 

1.02286 
1.04263 
1.04481 

+  0.00097 
+  0.00043 
+  0.00009 

+  0.00050 

November  7 
«         « 

11*  48'  A.M. 

0   21  P.M. 

5 
5 

Francis 

u 

0.97829 
0.97734 

3 
3 

Avery 

u 

0.97883 
0.97800 

+  0.00054 
+  0.00066 

+  0.00060 

November  4 

u           u 

«         7 
<c         « 

(£                « 

1*     5'  A.M. 
1     11       " 

11      9     " 
11    15     « 
11    31     « 

8 
8 
8 
8 
8 

Francis 
Haeffely 
Francis 

u 
« 

1.03882 
1.03701 
0.97501 
0.97579 
0.97530 

4 
4 
4 
4 
4 

Newell 

K 
M 

Avery 
Newell 

1.03940 
1.03881 
0.97677 
0.97761 
0.97700 

—  0.00058 
—  0.00180 
—  0.00176 
—  0.00182 
—  0'.00170 

—  0.00153 

November  7 

«           u 

2*  43'  P.M. 
30" 

5 
5 

Francis 
u 

0.80266 
0.80731 

1 
1 

Avery 
Newell 

0.80311 
0.80806 

+  0.00045 
+  0.00075 

+  0.00060 

November  7 

u           u 

4*  25'  P.M. 
4  41     « 

5 
5 

Francis 
u 

0.81346 
0.80972 

2 
2 

Avery 
Newell 

0.81432 
0.81079 

+  0.00086 
+  0.00107 

+  0.00096 

November  7 
«<        « 

3*  18'  P.M. 
3  .36     " 

8 
8 

Francis 

u 

0.80984 
0.81362 

4 
4 

Avery 
Newell 

0.81072 
0.81491 

—  0.00088 
—  0.00129 

—  0.00108 

EXPERIMENTS   ON   THE   FLOW   OF   WATER   OVER  WEIRS.  143 

176.     It  has  been  stated  (art.  164)  that  the  formula 

• 

%  (1.) 


is  applicable  only  to  a  weir  in  which  the  crest  is  horizontal.  Professor  James 
Thomson,*  of  Queen's  College,  Belfast,  has  deduced  from  formula  (1.)  a  formula  for 
the  discharge  over  symmetrical  triangular  notches  or  weirs,  figure  4,  plate  XVIII., 
viz. :  — 

Q2  —  2.664m  H^,  (2.) 

in  which 

Q2  —  the  discharge  in  cubic  feet  per  second. 

m  =  the  cotangent  of  the  inclination  of  the  crest  to  the  horizon,  on  each  side 

ED 
of  the  vertex  D,  equal  to  -p-™- 

Hz  =  the  depth  B  D  on  the  vertex  of  the  notch ;  the  line  ABC  being  the 
level  of  the  surface  of  the  water,  far  enough  from  the  notch,  to  be  un- 
affected by  the  curvature  caused  by  the  discharge. 

We  can  easily  deduce  from  (2.)  a  formula  for  the  case  in  which  the  crest  of  the 
weir  has  a  uniform  inclination  from  one  end  to  the  other. 

Formula  (2.)  gives  the  discharge  for   the  notch  ADC,  figure   4,   plate  XVIIL, 
in  which  A  B  =  C  B.     The  discharge  Q3  for  one  half  of  the  notch  A  B  D  is 

#3=  1.332m  #2  i  (3.) 

The  discharge  Q±  of  the  portion  of  the  notch  F  G  B  D  is  the  difference  of  the 
discharge  of  A  B  D  and  A  F  G.     Calling  F  B  =  L  and  F  G  =  Hz. 

Qi  =  1.332  m  H.%  —  1.332  m  H$, 
from  which  we  deduce 

Qt  =  1.332  m  (fff  -  -  Iff).  (4.) 

By  its  definition  m  =  -& ^ : 

-"2  -"3 

substituting  this  value  of  m  in  (4.),  we  have 

a  =  1.332    „  L  „  (fff  —  Iff}.  (5.) 

-"2  -"3     \  / 

*  Civil  Engineer  and  Architects'  Journal  for  April,  1863. 


144  EXPERIMENTS   ON  THE    FLOW   OF    WATER  OVER   WEIRS. 

Introducing  the  correction  for  the  end  contraction,  formula  (5.)  becomes 


- 

Qt  =  1.332  -  -jf—jj^-  (Of  -  Bfy  (6.) 


Formula  (6.)  is,  of  course,  applicable  only  to  weirs  of  the  same  section  in  the 
direction  of  the  flow,  as  formula  (1.),  from  which  it  is  deduced  (see  figure  3,  plate 
XVIII),  the  depth  at  one  end  of  the  weir  being  ff2  and  at  the  other  end  ffa.  When 
the  difference  in  the  depths  is  small,  relatively  to  the  mean  depth,  the  quantity 
computed  for  the  mean  depth  by  formula  (1.)  for  horizontal  crests  will  differ  but 
little  from  the  quantity  computed  by  formula  (6.),  as  will  be  seen  by  the  follow- 
ing examples  :  — 

Let  L  =  10,  and  the  mean  depth  on  the  weir  =  \  foot. 

By  the  formula  for  a  horizontal  crest,      .     .     .     Q  =  32.6340  cub.  ft.  per  sec. 

If  the  crest  is  0.01  foot  higher  at  one  end  than 

at  the  other,  by  formula  (6.)  ......     &  =  32.6340         «  « 

If  the  crest  is  0.1  foot  higher  at  one  end  than 

at  the  other,  by  formula  (6.)   ......     Q±  —  32.6442         "  « 

The  formula  for  the  discharge  of  a  weir,  deduced  from  the  theoretical  velocity 
of  water  issuing  from  an  orifice,  is 


00 

Q,  L,  and  H  having  the  same  signification  as  in  art.  164,  and  g  being  the  velocity 
acquired  by  a  body,  at  the  end  of  the  first  second  of  its  fall  in  a  vacuum,  which 
varies  with  the  latitude  of  the  place  and  its  height  above  the  level  of  the  sea. 

In  this  formula  L  H  represents  the  area  of  the  orifice  and  f  y/  2  g  H  the  mean 
velocity.  Applying  this  formula  to  a  weir  in  which  the  contraction  is  complete, 
both  at  the  ends  and  on  the  crest,  two  corrections  must  be  introduced;  that  for  the 
ends  amounting,  as  we  have  seen  (arts.  123,  124),  in  weirs  of  considerable  length  in 
proportion  to  the  depth  of  water  flowing  over,  to  a  diminution  of  the  length,  by 
a  quantity  depending  only  on  the  depth,  and  which  we  have  found  by  experiment 
(art.  156)  to  be  0.1  n  ff,  making  the  effective  area  of  the  weir  (L  —  0.1  n  H)  H. 

The  correction  for  the  contraction  on  the  crest  may  be  applied  in  the  form 
of  a  coefficient  m  of  the  velocity,  which  then  becomes  |  m  y/  2  g  H. 

Introducing  these  corrections,  formula  (1.)  becomes 

/ 

J.  (L) 


EXPERIMENTS   ON   THE    FLOW   OF    WATER  OVER   WEIRS.  145 

Taking  H  from  under  the  radical,  we  have 

Q  =  \m  \l^g  (L  —  0.1  n  H)  H%.  (2.) 

Formula  (2.)  is  identical  with  that  determined  by  experiment  and  given  in 
art.  164,  except  that  the  coefficient  3.33  is  replaced  by  f  m  y'  2  g.  In  order  that 
both  formulas  may  give  the  same  value  of  Q,  we  must  have 

|  m  y/T<7  =  3.33. 

Substituting   the  value   of  g  for  Lowell,  where  the   experiments  were   made,  we 

find 

m  =  0.6228. 

Substituting  this  value  of  m  in  (2.),  we  have 

Q  =  0.4152  <J~2g  (L  —  0.1  n  H)  H%,  (3.) 

by  which  formula  the  discharge  of  a  weir,  in  which  the  depth  flowing  over  is  not 
greater  than  one  third  of  the  length  and  the  contraction  complete,  may  be  com- 
puted for  any  latitude  and  height  above  the  sea,  by  introducing  the  corresponding 
value  of  g,  which  is  given  for  several  latitudes  and  heights  above  the  sea  in  a  table 
at  the  end  of  this  volume. 


19 


A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER  IN  OPEN  CANALS  OF  UNIFORM 
RECTANGULAR  SECTION,  AND  OF  SHORT  LENGTH. 


177.  THE  distribution  of  the  "Water  Power  at  Lowell  among  the  different  manu- 
facturing establishments,  in  accordance  with  the  rights  of  the  several  parties,  renders 
it  necessary  to  make  frequent  gaugings  of  the  quantities  of  water  drawn  by  them 
respectively.  In  all  the  leases  of  Water  Power  given  by  the  Proprietors  of  the  Locks 
and  Canals  on  Merrimack  River  there  is  the  following  provision :  — 

"  For  the  purpose  of  ascertaining  the  quantity  of  water  drawn  from  the  said  canals  or 
either  of  them  by  the  said  party  of  the  second  part  or  their  assigns,  the  said  Proprietors 
shall  have  the  right,  from  time  to  time,  as  they  may  desire,  by  their  duly  authorized  Agent, 
Engineer,  or  other  officer,  and  with  the  necessary  workmen  and  assistants,  to  enter  upon  the 
premises  of  the  said  party  of  the  second  part,  and  to  do  all  acts  (with  as  little  injury  as 
may  be)  necessary  or  proper  for  the  measuring  and  ascertaining  the  quantity  of  water  so 
drawn  as  aforesaid.  And  to  this  end  the  said  party  of  the  second  part  shall  render  all  need- 
ful and  proper  facilities ;  and  in  case  they  shall  suffer  any  loss  or  damage  by  the  acts  and 
doings  of  the  said  Proprietors  in  so  measuring  and  ascertaining  the  quantity  of  water  drawn 
as  aforesaid,  they  shall  be  entitled  to  compensation  therefor,  to  be  paid  by  the  said  Proprie- 
tors, the  amount  of  which  shall  be  ascertained  and  determined  by  arbitrators  appointed  and 
acting  according  to  the  -provisions  of  the  Agreement  of  1848  before  mentioned." 

From  the  nature  of  the  case,  it  is  necessary  to  make  the  gaugings  when  the 
manufacturing  operations  are  proceeding  in  the  usual  manner.  The  large  number  of 
persons  employed,  varying  from  five  hundred  in  the  smallest  establishment  to  more 
than  two  thousand  in  the  largest,  renders  any  interference  with  the  ordinary  course  of 
the  work  very  objectionable.  The  delicacy  of  many  of  the  operations  'also,  as  well  as 
the  large  pecuniary  interests  involved,  renders  such  establishments  extremely  sen- 
sitive to  any  interruption  to  their  normal  condition.  As  the  only  mode  of  avoiding 
frequent  and  troublesome  controversies  under  the  above  provision  in  the  leases,  the 
methods  adopted  for  gauging  the  quantity  of  water  drawn  have  been  limited  to 
such  as  could  be  applied  without  affecting  the  usual  course  of  operations  in  the 
manufacturing  establishments. 


A  METHOD  OF   GAUGING  THE   FLOW  OF  WATER  IN  OPEN  CANALS.         147 

It  will  readily  be  understood,  that  this  limitation  is  often  an  embarrassment  to 
the  Engineer  charged  with  the  duty  of  making  the  gaugings.  The  simple  and 
exact  method  of  the  weir  is  rarely  applicable,  as  it  would,  in  most  cases,  detract 
materially  from  the  effective  fall  operating  upon  the  water-wheels.  Seldom  less  than 
two  feet  fall  would  be  required  for  this  purpose,  and  the  cases  are  exceptional  where 
such  an  amount  of  fall  could  be  taken  from  that  usually  used,  without  causing  in- 
terruption to  the  manufacturing  operations  dependent  upon  the  power.  The  same 
objection  applies  to  gaugings  by  means  of  apertures  of  any  kind,  excepting,  however, 
the  apertures  by  which  the  water  is  applied  directly  to  the  water-wheels ;  such 
apertures  are,  however,  constructed  more  with  reference  to  the  requirements  of  the 
water-wheel  as  a  motor  than  to  the  making  of  accurate  gaugings  of  the  quantities 
of  water  passing  through  them,  and  if  the  Engineer  attempts  to  compute  the  flow 
through  them  by  the  known  laws  of  hydraulics,  he  generally  finds  himself  beset  with 
such  difficulties  and  uncertainties  as  to  prevent  any  confidence  in  the  results,  except 
as  approximations. 

178.  In  the  gaugings  at  Lowell   the  weir  is   sometimes,  although  rarely,  admis- 
sible;  gaugings  by  means  of  the  apertures  by  which  the  water  is  applied  directly  to 
the  water-wheels  are  more  frequent,  but,  as  a  rule,  only  where  experiments  have  been 
made  on  the  discharge  of  the  particular  water-wheel  or  one  of  the  same  form.     The 
water   drawn   by  the    Suffolk  Manufacturing  Company  and   by   the   Tremont  Mills   is 
now   ascertained   in   this   manner,   all   the  water   drawn   by  them  being   used  on  tur- 
bines of  the   same   form   and  dimensions  as  that  experimented  upon  at  the  Tremont 
Mills  in  the   year   1851 ;   an   account  of  the   experiments   on  which   is  given   in   the 
first  part  of  this  work.     A  similar  course  is  adopted  at   two   other   establishments   in 
which  the  water  is   used   upon   turbines;  in  both   of  these  cases  the  discharge  of  a 
turbine   of  each   different  pattern   has  been   determined   by   means   of   weirs,   under 
various   circumstances  as  to  height   of  gate,   velocity   of  rotation,  etc.,  and   from   the 
data   thus   obtained   tables   have   been   prepared;   by   means   of  which   the   discharge 
is    at    any    time    readily   obtained,   from   the    observed   height   of    gate,   velocity   of 
rotation  of  wheel  and  the  fall.     Care  must  be  taken,  however,  that  the  wheels  are  in 
good  running  order  when  the  observations  are  made. 

179.  Generally,  the  water   used  at  a  manufacturing  establishment  is   all  drawn 
from    the   same    canal   or  watercourse,  at  several   points  on   the  same  bank,  through 
covered   penstocks ;   and   from   each  of  these  the  water  is  delivered   to  one  or  more 
water-wheels,   and   in   some    instances   to   several    smaller   apertures,    where   water   is 
drawn   for   other   purposes   than  for  that  of  furnishing  mechanical  power.     To  gauge 
the  quantity  of  water  drawn   simultaneously  at  all  these  points  would  be  a  work  of 
much   difficulty,  under    the   most  favorable    circumstances,  and  when   hampered  with 


148  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

the  limitation  that  it  must  be  done  without  interference  with  the  ordinary  operations 
of  ..the  establishment,  it  becomes  impracticable.  The  difficulties  mainly  disappear, 
however,  if  the  gauge  can  be  made  in  gross  before  or  after  the  water  enters  the 
establishment;  and  it  has  been  a  matter  of  great  interest  here  to  devise  and  perfect 
methods  by  which  this  could  be  satisfactorily  done. 

180.  In   the   year  1830,  the  quantity  of  water  drawn  at  one  of  the  cotton-mills 
of  the  Hamilton  Manufacturing  Company  was   measured  by  means   of  a   gauge-wheel 
15  feet  in   length  and  19.25  feet  in  diameter,  which  operated  in  a  manner  somewhat 
similar    to  an  ordinary  wet  gas-meter.      The  gauge-wheel  was    placed  in  the  tail-race 
of  the   mill,  where  all   the  water  used  in  it  was  discharged.     The  quantity  of  water 
thus  gauged  was  about  90  cubic  feet  per  second.* 

181.  In  the   year   1841,   Messrs.   James   F.   Baldwin,   George   W.   Whistler,  and 
Charles  S.  Storrow,  three  eminent  engineers,  were  appointed  Commissioners  to  deter- 
mine the  quantities  of  water  drawn  from  the  canals  of  the    Proprietors  of  the  Locks 
and  Canals   on   Merrimack  Biver,   by  the    several  manufacturing    companies   at   Low- 
ell.    The  following  extracts   are   from   their  reports,  which  have  never  been  printed. 

Extract  from  first  report,  dated  October  8,  1841  :  — 

"  Upon  considering  how  we  should  best  effect  the  object  in  view,  various  methods  occurred 
to  us,  given  in  the  books  on  the  subject,  by  which  the  quantity  of  water  passing  through 
a  canal  is  deduced  by  calculation  from  elements  easily  measured,  such  as  the  velocity  at  the 
surface,  the  slope  and  the  several  dimensions  of  the  canal.  For  many  purposes  these  rules 
would  be  sufficient,  and  if  applied  herve  would  give  us  an  approximation.  The  experiments  on 
which  they  depend  having  been,  however,  generally  conducted  on  a  small  scale,  and  not  always 
consistent  with  each  other,  we  did  not  feel  willing  to  -  trust  to  their  decision  interests  so  im- 
portant as  those  involved  in  the  question  before  us.  The  application  of  such  rules  would 
occasion,  it  is  true,  but  little  expense,  but  for  that  little  expense  would  furnish  only  very 
imperfect  information. 

"  It  appeared  to  us,  therefore,  that  the  only  satisfactory  mode  of  proceeding  was  to  make 
a  direct  and  positive  measurement  of  the  quantity  of  water  flowing  through  the  Merrimack 
and  Western  Canals,  which  afford  greater  facilities  for  the  purpose  than  the  others,  and  by 
that  means  to  obtain  not  only  the  true  quantity  passing  there  at  the  present  time,  but  to 
test  a  rule  of  easy  application  to  the  other  canals,  by  which  the  quantity  which  they  con- 
vey can  be  ascertained  without  the  expense  of  a  similar  measurement,  and  by  which  also 
the  quantities  passing  in  any  of  the  canals  may  at  any  future  time  be  very  easily  deter- 
mined. 

"  In  pursuance  of  this  plan  we  selected  a  convenient  spot  in  the  Western  Canal,  near 
the  Tremont  and  Suffolk  Mills,  where  it  is  about  twenty-nine  feet  wide  and  eight  feet  deep. 

*  See  Journal  of  the  Franklin  Institute  of  Pennsylvania,  Vol.  XI.  2d  .Series,  1833. 


IN   OPEN   CANALS   OF   UNIFORM   RECTANGULAR   SECTION.  149 

We  there  excavated  the  earth  from  the  sides  and  formed  a  basin  about  eighty  feet  across  in 
the  widest  place,  and  raised  the  bottom  so  as  to  leave  the  depth  only  about  four  feet  six 
inches.  We  there  placed  across  the  canal  seven  paddle-wheels,  sixteen  feet  in  diameter  and 
ten  feet  long  each,  with  narrow  and  solid  piers  between  them,  and  coupled  the  shafts,  to 
make  them  all  revolve  together  as  one  piece.  These  wheels  were  made  with  great  care,  and 
were  so  accurately  fitted  as  to  run  within  about  a  quarter  of  an  inch  of  the  apron  or  floor 
below  them,  and  the  piers  at  the  sides,  thus  filling,  as  nearly  as  possible  in  practice,  the 
whole  of  the  seven  spaces  included  between  the  piers.  By  driving  sheet  piling  across  the 
head  of  the  apron  and  into  the  banks,  we  obliged  all  the  water  of  the  canal  to  pass  between 
the  piers  and  drive  the  wheels.  The  apron  was  formed  of  timbers  cut  to  a  true  sweep,  cor- 
responding to  the  circle  described  by  the  bottom  of  the  floats,  and  was  of  sufficient  length, 
in  the  direction  of  the  current,  for  one  float  to  enter  it  at  the  upper  side  before  the  preceding 
float  had  left  it  on  the  lower.  If  the  wheel,  therefore,  accurately  fitted  the  apron  and  the  piers, 
it  is  evident  that  when  two  successive  floats  were  over  the  apron  at  the  same  time,  the  body 
of  water  included  in  the  space  between  them  and  the  apron  (which  we  call  a  bucket)  was 
cut  off  from  the  rest  and  passed  by  itself;  and  as  the  wheels  revolved,  all  the  water  of 
the  canal  could  only  pass  in  this  manner  by  successive  buckets  full.  A  clock  fixed  upon  the 
end  of  the  shaft  showed  the  number  of  revolutions  of  the  wheel,  and  consequently  the  number 
of  buckets  passed  in  any  given  time.  If  we,  therefore,  could  tell  just  the  quantity  of  water 
contained  in  a  bucket,  or  between  the  two  floats  when  over  the  apron,  we  had  simply  to 
multiply  it  by  the  number  of  buckets  passed,  and  we  had  at  once  the  whole  quantity  of 
water  for  the  given  time. 

"  To  ascertain  the  quantity  of  water  in  a  bucket,  knowing  all  the  dimensions  of  our 
wheels,  we  only  needed  to  get  the  depth  of  the  water  above  the  apron.  This  would  vary 
according  to  the  variations  in  the  level  of  the  canal,  and  was  observed  and  noted  every  five 
minutes  during  the  whole  period  of  our  experiments,  by  means  of  gauges  fixed  upon  some 
of  the  piers  and  upon  the  floats  themselves. 

"  The  foregoing  description  shows  the  manner  in  which  we  obtained  a  direct  measure- 
ment in  the  Western  Canal.  To  obtain  the  other  object,  that  is,  to  test  a  simpler  mode  of 
measuring,  to  be  used  in  the  other  canals  and  in  future  in  this,  we  placed  at  some  distance 
above  the  wheels  a  flume  or  wooden  trunk  of  a  section  nearly  equal  to  that  of  the  canal, 
to  the  bottom  and  sides  of  which  it  thus  formed  a  lining.  The  bottom  of  the  flume  very 
nearly  coincided  with  the  bottom  of  the  canal,  and  was  covered,  as  well  as  the  sides,  with 
plank  carefully  jointed  so  as  to  form  a  smooth  and  even  surface.  The  length  of  the  flume 
was  150  feet ;  the  width,  27.22  feet ;  the  water  was  generally  about  eight  feet  deep.  Being 
of  such  a  size  it  produced  no  sensible  disturbance  in  the  flow  of  the  water,  and  gave  us 
means  of  accurately  measuring  the  dimensions  of  the  stream  as  it  passed  through  it.  Sheet 
piling  was  of  course  driven  at  its  upper  end,  so  as  to  throw  all  the  water  through  it.  Its 
lower  end  was  151  feet  distant  from  the  wheels. 

"  Simultaneously  then  with  the  observations  which  we  made  on  the  quantity  at  the  wheels, 
we  carried  on  another  series  at  the  flume,  through  which,  of  course,  the  same  quantity  was 
passing.  We  carefully  noted,  about  once  in  2.5  minutes,  the  depth  of  water  and  the  number 


150  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

of  seconds  in  which  a  small  float,  placed  in  the  centre  of  the  stream,  at  the  surface,  passed 
through  a  space  of  130  feet  in  length,  measured  on  the  flume.  The  width  of  the  flume  and 
the  depth  being  known,  we  knew,  therefore,  at  the  moment  of  each  observation,  the  section 
of  the  stream  and  the  velocity  of  the  water  at  the  surface  in  the  centre. 

"  It  was  long  since  ascertained  by  experiments  made  on  a  small  scale,  that  a  certain 
ratio  exists  between  the  surface  velocity  thus  measured  and  the  mean  velocity,  or  that  which, 
multiplied  by  the  section  of  the  stream,  gives  the  true  discharge ;  and  as  in  the  present  case 
we  knew  by  the  wheels  the  true  quantity  passing,  we  were  able  to  test  this  simple  rule,  and 
see  how  much  it  should  be  altered  and  corrected,  if  at  all,  in  order  to  give  accurate  results 
with  bodies  of  water  so  very  much  larger  than  those  hitherto  experimented  upon. 

"  It  is  this  rule,  thus  corrected,  and  further  tested  by  a  similar  course  of  experiments 
with  wheels  and  a  flume  in  the  Merrimack  Canal,  that  we  propose  to  use  for  the  measure- 
ments in  the  other  canals  at  Lowell.  The  expense  of  erecting  the  wheels  is  great,  and  they 
could  not  of  course  be  left  permanently  in  the  canals.  The  flumes  cost  much  less,  interfere 
neither  with  the  navigation  nor  with  the  passage  of  the  water,  and  are  intended  to  remain 
in  place  as  long  as  may  be  desired.  At  any  future  time,  therefore,  it  would  only  be  neces- 
sary to  measure  the  depth  of  water  in  them,  and  the  surface  velocity,  and  deduce  at  once, 
by  the  rule,  the  quantity  passing  through  them." 

Extract  from  the  third  and  final  report  of  the  Commissioners,  dated  Decem- 
ber 17,  1842:  — 

"  In  our  first  report,  made  in  October,  1841,  we  stated  that  we  hoped  to  make  our 
experiments  serve,  not  only  to  give  us  a  measurement  of  the  quantity  of  water  passing  at  the 
present  moment,  but  to  test  and  verify  a  method  or  rule  for  finding  the  quantity  at  a  future 
time,  without  the  necessity  of  the  heavy  expenditure  now  incurred.  As  the  result  of  our 
labor,  we  recommend  for  future  use  the  following  rule  for  measuring  the  quantities  passing 
through  the  open  flumes  which  we  have  erected  in  the  various  canals  leading  the  water  to 
the  mills. 

"  Multiply  the  depth  in  feet  by  the  width  in  feet  of  the  stream  where  it  passes  through 
the  flume,  and  the  product  by  the  velocity  at  the  surface,  in  feet  per  second ;  this  velocity 
being  found  by  noting  the  time  in  which  a  small  float,  just  immersed  in  the  quickest  part 
of  the  stream,  passes  through  a  given  distance.  Multiply  the  quantity  then  found  by 

0.847  in  the  Western  Canal, 

0.814  in  the  Merrimack  Canal, 

0.835  in  the  Hamilton  and  Appleton  Canal, 

0.830  in  the  Eastern  Canal, 

0.810  in  the  Lowell  Canal, 

and  the  result  is  the  number  of  cubic  feet  per  second  passing  through  the  flume. 

"  Should  there  be  in  future  any  great  change  in  the  velocity  with  which  the  water  passes 
through  the  canals,  these  constant  numbers  or  multipliers  would  require  some  alteration.  We 


IN  OPEN  CANALS  OF  UNIFORM   RECTANGULAR  SECTION. 

* 

may  state,  in   general  terms,    that   these   numbers   should   be   increased  in   case   of   a  greatly 

increased  velocity,  and  diminished  for  a  velocity  greatly  diminished.  Within  the  limits,  how- 
ever, of  the  ordinary  variations  in  the  canals,  as  they  are  now  used,  they  may  be  considered 
as  fixed  and  constant  for  the  same  canal. 

"  To  show  the  application  of  the  rule,  and  how  far  it  can  be  relied  on  for  accuracy, 
we  refer  to  the  annexed  table  marked  A.  In  that  table  the  numbers  of  column  6  show  the 
depth  of  the  water  in  the  flume,  which  in  the  first  experiment  was  8.03  feet.  Column  7 
shows  the  number  of  seconds  in  which  the  float  ran  130  feet,  which  in  that  case  was  41.47 
seconds.  Column  8  gives  the  surface  velocity  per  second,  3.135  feet  (in  experiment  1),  found 
by  dividing  130  feet,  the  distance  run,  by  41.47  seconds,  the  time  occupied.  Multiplying  the 
depth,  8.03,  by  the  width,  27.22,  which  gives  218.58,  and  this  product  by  the  velocity,  3.135, 
we  find  the  quantity,  685.25,  given  in  column  9.  This  quantity,  we  may  observe,  would  be 
the  true  quantity,  if  the  velocity  of  every  portion  of  the  stream  was  the  same  as  the  surface 
velocity.  Multiplying  685.25  by  0.847,  which  is  the  constant  multiplier  for  this  canal,  we 
obtain  580.41  in  column  11  for  the  number  of  cubic  feet  per  second,  according  to  calculation. 
Actual  measurement  at  the  gauge-wheels  gives  us  586.69  in  column  12,  for  the  true  quantity. 
Calculation,  therefore,  gives  in  this  case  6.28  cubic  feet  less  than  measurement,  as  shown  in 
column  13 ;  and  6.28,  the  amount  of  error,  is  only  about  one  per  cent  of  the  true  quantity 
586.69,  or,  more  exactly,  0.0107  of  586.69,  as  shown  in  column  14. 

"  We  refer  to  our  report  made  in  October,  1841,  for  a  description  of  the  manner  in 
which  our  experiments,  made  simultaneously  at  the  flumes  and  at  the  gauge-wheels,  were 
conducted.  The  experiments  then  described  as  made  in  the  Western  Canal  were  repeated 
this  year,  in  exactly  the  same  manner,  in  the  Merrimack  Canal,  where  the  velocity  was  about 
two  thirds  as  great  as  in  the  other. 

"Table  A  shows  the  results  of  all  the  experiments  in  both  canals.  Comparing  the  cal- 
culated quantities  in  each  experiment,  given  in  column  11,  with  the  measured  quantities  in 
column  12,  it  will  be  seen,  that  in  the  Western  Canal  the  greatest  difference  between  the 
calculation  and  the  measurement  is  about  one  per  cent,  and  the  mean  difference  in  the 
experiments  in  that  canal  is  something  less  than  one  half  per  cent.  In  the  Merrimack  Canal 
there  is  one  experiment,  the  19th,  in  which  the  proportional  difference  was  between  three  and 
four  per  cent.  In  all  the  other  experiments  it  was  less  than  two  per  cent,  and  the  mean 
difference  of  the  whole  was  about  one  per  cent.  The  experiment,  No.  19,  in  which  the 
greatest  difference  occurred,  was  manifestly  made  under  less  favorable  circumstances  than  any 
of  the  rest,  there  having  been  a  great  irregularity  in  the  depth  of  the  water,  as  shown  by 
the  note  in  the  table.  In  estimating  the  degree  of  accuracy  which  the  flume  rule  will  give 
us,  it  would  perhaps  be  proper  to  throw  out  this  experiment. 

"In  addition  to  Table  A,  which  contains  all  our  experiments,  and  is  the  one  from  which 
the  constant  coefficients  are  determined,  we  have  given  two  other  tables,  B  and  C,  which 
contain  the  same  experiments  divided  into  short  periods,  with  two  others,  on  the  accuracy  of 
which  we  could  not  place  quite  so  much  reliance.  These  tables  show,  of  course,  greater 
variations  between  calculation  and  measurement  than  the  other,  and  could  hardly  fail  to  do 
so ;  just  as  a  single  observation  is  less  accurate  than  the  mean  of  several  made  with  equal 


152 


A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 


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154  A  METHOD  OF   GAUGING  THE  FLOW  OF  WATER 

care.  Little  inaccuracies  are  unavoidable  in  measuring  the  time  and  depth  ;  and  an  occasional 
eddy  or  cross  current,  or  other  accidental  cause,  may  vary  the  observed  velocity,  and  con- 
sequently the  calculated  discharge  ;  and  such  inaccuracies  have  less  influence  on  the  result 
of  observations  long  continued  than  of  a  smaller  number,  taken  in  a  shorter  time.  Still, 
the  comparison  of  calculation  and  measurement  given  in  these  two  tables  shows  that  the 
mean  difference,  on  the  whole,  is  but  from  one  to  two  per  cent,  sometimes  in  excess  and 
sometimes  in  deficiency. 

"  It  may  be  of  some  interest  to  compare  the  accuracy  shown  by  our  tables  with  that 
shown  in  the  table  of  M.  De  Prony,  a  French  engineer  of  the  highest  reputation,  whose  rules 
have  generally  been  adopted  in  France.  To  determine  the  relation  between  the  surface  and 
the  mean  velocity,  he  used  seventeen  experiments  made  in  France  by  Du  Buat,  on  a  small 
scale,  in  little  wooden  troughs  about  eighteen  inches  wide,  with  depths  of  from  two  to  ten 
inches,  and  velocities  varying  from  six  inches  per  second  to  four  feet  and  three  inches  per 
second.  He  gives  0.816  as  the  decimal  by  which  to  multiply  the  surface  velocity  in  order 
to  reduce  it  to  the  mean  velocity.  Comparing  his  quantities  so  calculated  for  these  seventeen 
experiments  with  the  quantities  actually  measured,  he  finds  proportional  differences  amount- 
ing, for  the  mean  of  the  whole,  to  a  little  less  than  five  per  cent,  the  greatest  difference 
being  about  fourteen  per  cent.  As  his  velocities  varied  very  much,  he  found  that  he  could 
calculate  the  discharge  more  accurately  by  varying  the  number  0.816,  taking  a  smaller  number 
for  low  velocities  and  a  larger  number  for  high  velocities.  Calculating  the  quantities  by  his 
most  exact  rule,  in  which  due  influence  is  given  to  the  variation  of  velocities,  he  found 
results  differing  from  measurement  about  three  per  cent  on  an  average,  after  throwing  out 
two  of  the  seventeen  experiments  which  showed  a  much  greater  difference,  and  which  he 
considered  less  satisfactory  than  the  rest.  He  remarks,  that,  as  his  most  correct  rule  gives  a 
result  which  is  within  about  one  thirtieth  of  the  truth,  it  ought  to  be  considered  as  more 
than  sufficiently  correct  for  practical  purposes.  Our  own  table  A  shows  a  much  closer  cor- 
respondence between  calculation  and  measurement,  as  it  naturally  should  do,  because  the 
variations  of  velocity  and  change  of  circumstances  in  our  experiments  in  each  canal  were 
much  less  than  they  were  in  the  experiments  made  in  France.  It  is  a  remarkable  circum- 
stance that  the  rules  of  M.  De  Prony  should  apply  as  closely  as  they  do  to  our  case, 
where  the  section  of  the  stream  is  400  or  500  times  as  large  as  it  was  in  his  experi- 
ments." * 

*  Prony's  more  correct  formula,  reduced  to  the  English  foot  as  the  unit,  is 

7.78188) 


V+  10.34508  ' 

in  which  V=  the  surface  velocity  in  the  middle  of  the  stream,  and  v  =  the  mean  velocity.  (Storrow  on 
Water  -Works,  p.  96.) 

Putting  the  constants  in  (1.)  equal  to  A  and  B,  we  can  determine  their  values  from  the  experiments 
in  table  A. 

Taking  the  mean  values,  we  have  at  the  Western  Canal  V=  3.205  and  v  =  0.847  X  3.205,  and  at 
the  Merrimack  Canal  V=  2.138  and  v  =  0.814  X  2.138. 


IN  OPEN  CANALS  OF  UNIFORM  RECTANGULAR  SECTION.  155 

182.  In  connection  with  the  measurement  of  the  quantities  of  water  drawn  by 
the  several  manufacturing  corporations  at  Lowell,  undertaken  in  the  year  1852,  the 
method  of  gauging  in  measuring  flumes  placed  in  the  feeding  canals  naturally  re- 
ceived much  attention.  The  flumes  constructed  in  the  years  1841  and  1842  were 
generally  in  good  order,  and  had  been  used  at  intervals  as  originally  intended. 
Serious  doubts,  however,  arose,  as  to  whether  it  was  safe  to  apply  the  rules  deduced 
from  the  experiments  at  the  flumes  in  the  Western  and  Merrimack  canals,  to  those 
in  some  of  the  other  canals.  In  both  the  Western  and  Merrimack  canals  the  water 
at  its  arrival  at  the  flumes  had  passed  through  more  than  a  thousand  feet  of 
canal,  of  nearly  uniform  section,  without  having  any  part  of  its  volume  abstracted. 
In  the  Western  canal,  the  nearest  bend  on  the  up-stream  side  of  the  flume  was 
about  six  hundred  feet  distant,  and  in  the  Merrimack  canal  about  two  thousand 
feet.  It  was  thought  that,  in  the  passage  of  the  water  to  the  flumes,  under  these 
circumstances,  the  velocities  in  different  parts  of  the  section  would  become  adjusted, 
according  to  the  natural  laws  governing  the  flow  of  water  in  regular  channels  of 
great  length,  and  that  while  it  might  be  safe  to  compute  the  flow  in  other  flumes, 
similarly  situated  as  to  the  approaches,  from  the  observed  surface  velocity,  it  would 
not  be  so  in  other  cases,  where  the  length  of  canal,  immediately  above  the  flume, 
in  which  the  direction,  section,  and  velocity  were  nearly  the  same  as  in  the  flume, 
was  too  short  to  allow  of  such  an  adjustment  of  velocities  in  different  parts  of 
the  section.  All  the  other  measuring  flumes  were  much  less  favorably  situated 
in  this  respect  than  those  in  the  Western  and  Merrimack  canals;  one  of  them 
designed  to  gauge  the  largest  quantity,  being  immediately  below  the  entrance  to 
the  canal,  and  the  others  were  liable  to  be  affected  by  bends,  and  other  irreg- 
ularities, at  short  distances  above  them.  The  difficulty  lay  in  the  uncertainty  as 
to  whether  the  velocities  at  the  surface  and  at  other  parts  of  the  section  would 

Substituting  these  values  in  (1.),  we  have 


from  which  two  equations  we  find  A  =  1.889  and  B  =  2.809,  and  the  formula  becomes 

_  F(F+ 1.889) 


(2.) 


F  +  2.809  ' 

Formulas  (1.)  and  (2.)  appear  to  differ  very  much,  but  it  will  be  found  that  at  ordinary  velocities  they 
give  values  of  v  which  differ  but  little.  In  figure  1,  plate  XVIII.,  the  line  ABC  represents  the  values 
of  v  by  formula  (1.),  and  the  line  A  B  D  the  values  by  formula  (2.).  Both  formulas  give  the  same  value 
of  v  when  V=  1.41,  corresponding  to  the  intersection  of  the  two  lines  at  B. 


156  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

bear  the  same  relations  to  each  other  under  different  circumstances  as  to  the 
approaches;  there  were  strong  reasons  for  believing  that  they  would  not,  and  that 
consequently,  the  quantity  computed  by  the  rules  for  deducing  it  from  the  surface 
velocity  would  be  liable  to  errors  of  such  magnitude  as  to  render  the  results 
valueless. 

183.  No  substitute  for  the  method  of  gauging  in  the  flumes  could  be  devised, 
and  the  proper  course  appeared  to  be  to  adopt  some  method  of  arriving  at  the 
mean  velocity  which  should  not  be  open  to  the  objections  urged  to  that  of 
deducing  it  from  the  surface  velocity.  What  appeared  to  be  required  was  a  correct 
and  convenient  method  of  taking  into  account  the  velocities  in  every  part  of  the 
section.  There  are  several  well-known  methods  designed  to  accomplish  this  result. 
Woltman's  mill,  or  tachometer,  has  been  much  used  for  this  purpose,  but,  to  insure 
correct  results,  its  application  is  one  of  much  delicacy,  and  in  our  large  channels 
would  require  much  time.  Submerged  floats,  Pitot's  tube,  the  hydrometric  pendulum, 
and  many  other  contrivances  are  described  in  the  books  on  hydraulics.  The  most 
promising  appeared  to  be  that  of  obtaining  the  mean  velocity  by  means  of  light 
rods  or  staves  loaded  at  one  end  so  that  they  would  float  vertically,  or  nearly 
so,  and  extend  nearly  to  the  bottom  of  the  channel.  The  advantages  of  this  method 
were  suggested  long  since.  The  following  extract  is  from  a  paper  on  rivers  and 
canals,  by  T.  H.  Mann,  read  before  the  Royal  Society  of  London,  and  printed  in 
their  Transactions  for  the  year  1779 :  — 

"  The  best  and  most  simple  method  of  measuring  the  velocity  of  the  current  of  a  rivei 
or  open  canal,  .that  I  know  of,  is  the  following :  — 

"  Take  a  cylindrical  piece  of  dry,  light  wood,  and  of  a  length  something  less  than  the 
depth  of  the  water  in  the  river  ;  round  one  end  of  it  let  there  be  suspended  as  many 
small  weights  as  may  be  necessary  to  keep  up  the  cylinder  in  a  perpendicular  situation  in 
the  water,  and  in  such  a  manner  that  the  other  end  of  it  may  just  appear  above  the  sur- 
face of  the  water.  Fix  to  the  centre  of  that  end  which  appears  above  water  a  small  and 
straight  rod,  precisely  in  the  direction  of  the  cylinder's  axis ;  to  the  end,  that  when  the  in 
strumont  is  suspended  in  the  water,  the  deviations  of  the  rod  from  a  perpendicularity  to  the 
surface  of  it  may  indicate  which  end  of  the  cylinder  advances  the  fastest,  whereby  may  be 
discovered  the  different  velocities  of  the  water  at  different  depths ;  for  if  the  rod  inclines 
forwards  according  to  the  direction  of  the  current,  it  is  a  proof  that  the  surface  of  the 
water  has  the  greatest  velocity ;  but  if  it  inclines  back,  it  shows  that  the  swiftest  current 
is  at  the  bottom ;  if  it  remains  perpendicular,  it  is  a  sign  that  the  velocities  at  the  surface 
and  bottom  are  equal. 

"  This  instrument  being  placed  in  the  current  of  a  river  or  canal  receives  all  the  percus- 
sions of  the  water  throughout  the  whole  depth,  and  will  have  an  equal  velocity  with  that  of  the 
whole  current  from  the  surface  to  the  bottom  at  the  place  where  it  is  put  in,  and  by  that  means 


IN   OPEN   CANALS   OF   UNIFORM    RECTANGULAR   SECTION.  157 

may   be   found,   both   with    ease    and    exactness,   the   mean   velocity   of    that   part   of  the    river 
for   any   determinate   distance   and   time. 

"  But  to  obtain  the  mean  velocity  of  the  whole  section  of  the  river,  the  instrument 
must  be  put  successively  both  in  the  middle  and  towards  the  sides,  because  the  velocities 
at  those  places  are  often  very  different  from  each  other.  Having  by  this  means  found  the 
difference  of  time  required  for  the  currents  to  run  over  an  equal  space;  or,  the  different  dis- 
tances run  over  in  equal  times,  the  mean  proportional  of  all  these  trials,  which  is  found  by 
dividing  the  common  sum  of  them  all  by  the  number  of  trials,  will  be  the  mean  velocity  of 
the  river  or  canal." 

Mann  does  not  claim  to  have  been  the  first  to  propose  this  method,  and  it 
is  probably  to  be  found  in  the  works  of  some  of  the  older  hydraulicians.  It  is 
frequently  mentioned  by  more  modern  writers ;  generally,  however,  as  one  of  the 
modes  which  have  been  proposed,  but  without  much  stress  being  laid  upon  it,  as 
being  a  convenient  and  accurate  method.  BufTon  gauged  the  Tiber  by  this  method, 
using  for  floats  small  bundles  of  rods,  so  loaded  at  one  end  as  to  float  almost 
vertically,  and  extending  from  the  surface  nearly  to  the  bottom.  Krayenhoff*  made 
some  use  of  it  in  gauging  rivers  in  Holland,  previous  to  the  year  1813 ;  but  in  apply- 
ing it  to  natural  watercourses  the  irregularities  in  the  depth  must  often  present 
difficulties,  not  met  with  in  rectangular  channels  of  uniform  section. 

184.  This  method  of  obtaining  the  mean  velocity  of  water  flowing  in  open 
channels,  not  being  that  commonly  used  by  engineers,  or  given  by  writers  of 
authority  on  the  subject  as  an  accurate  and  established  method,  it  was  necessary, 
at  its  first  introduction  here,  —  large  pecuniary  interests  being  involved,  —  to  prove 
its  accuracy,  or  at  least  to  ascertain  within  what  limits  of  error  it  could  be 
applied.  Accordingly,  in  the  year  1852,  some  direct  comparisons  were  made  be- 
tween the  results  obtained  by  gauging  the  flow  through  rectangular  channels,  in 
which  the  mean  velocity  was  measured  by  means  of  loaded  tubes,  and  by  gauging 
the  same  volume  of  water  by  means  of  weirs;  the  formula  for  computing  the  flow 
over  weirs  having  been  determined  by  experiments  on  a  suitable  scale.  These 
comparisons  are  described  in  the  first  edition  of  this  work.  They  indicate  a  close 
correspondence  in  the  results  arrived  at  by  the  two  methods;  as  might  be  expected, 
however,  the  quantity  deduced  from  the  mean  velocity  of  the  tubes  was,  generally, 
a  little  in  excess  of  the  mean  velocity  as  deduced  from  the  gauge  of  the  same 
volume  of  water  at  the  weirs;  the  greatest  difference  being  in  the  comparisons  in 

*  Recueil  des  observations  Hydrographiques  et  Topographiques  faites  en  Hollande,  par  C.  R.  T.  KEAYEN- 
HOFF.  Amsterdam,  1813. 

The  floats  used  by  Krayenhoff  were  wooden  poles,  loaded  with  lead  at  the  bottom,  and  buoyed  up  by  copper 
floats  at  the  surface  of  the  water. 


158  A  METHOD  OF   GAUGING  THE   FLOW  OF  WATER 

which  the  shortest  tubes  were  used,  the  excess  in  this  case,  however,  being  only 
about  four  per  cent.  These  comparisons  furnished  the  means  of  making  corrections 
of  the  flume  "measurements,  (by  which  term  is  to  be  understood  the  product  of  the 
mean  velocity  of  the  tubes  into  the  section,)  in  order  that,  the  results  might  be 
substantially  the  same  as  would  be  given  by  weir  measurements ;  and  also  estab- 
lished, beyond  question,  that  the  method  could  be  relied  upon,  when  applied  under 
favorable  circumstances,  to  give  results  sufficiently  near  the  truth  to  meet  the 
practical  requirements  of  all  the  parties  in  interest. 

The  experiments  of  1852  were  not  sufficiently  numerous  and  varied  to  afford 
the  data  for  a  formula  of  correction  of  general  application,  and  arrangements  having 
been  subsequently  made  between  the  lessors  and  the  lessees,  which  involved  more 
frequent  and  more  accurate  gaugings,  it  was  deemed  expedient  to  perfect  the 
method  as  far  as  practicable;  and  also  to  ascertain  the  extent  to  which  we  were 
liable  to  err  in  applying  the  method  in  some  peculiar  circumstances,  such  as  high 
winds,  or  with  irregular  currents  and  eddies  in  the  measuring  flumes.  Accord- 
ingly, in  the  year  1856,  an  extensive  series  of  experiments  was  made  for  these 
purposes,  an  account  of  which  is  given  below. 

185.  In   long   straight   channels,   in   which   the    section   occupied   by   the    water 
is   uniform,   and    the    quantity   of   water   flowing    is   constant,   at  a   distance,   greater 
or  less,   from    the   place   where    the   water   is   admitted,   a   certain   relation   is   estab- 
lished  between   the   slope   of  the    surface    and   the    mean   velocity;  and  also  between 
the   velocities   of    the   water   in   different  parts   of    the   section;    that   is,   the   regime 
is   established,   and   the   stream   is    said   to   be   in   a   state    of    uniform  motion.      The 
comparative   velocities   at   different    depths,   in    any   vertical    plane   which   is   parallel 
with   the    direction   of    the    current,   are    called   the    scale   of  velocities.      Most   of  the 
rules   given    by   writers    on    hydraulics  for   the   motion   of   water  in    open   channels 
are   for   the    case    of  uniform   motion. 

It  is  generally  assumed  that  the  resistance  to  the  motion  of  water  all  pro- 
ceeds from  the  bed,  by  which  is  meant  the  bottom  and  sides  of  the  channel, 
and  that  the  maximum  velocity  in  symmetrical  channels  of  the  usual  forms  is  at 
the  surface,  and  in  the  middle  of  the  stream. 

186.  When   the   air  in   contact  with   the   surface  of  water,  flowing   in   an  open 
channel,   is  moving   in  the  same  direction,  and  with  the  same  velocity,  as  the  surface 
of  the   water,   it   is   clear   that  it   can   have   no    effect   on  the  motion   of  the  water ; 
but    such    exact    conformity   in    the    motion    of   the    air    and   water   is   uncommon ; 
ordinarily,   the    air    has   some   motion   relatively   to   that    of    the    water,   and    either 
retards  or    accelerates    the   velocity   of   the    surface.     That  the   air  may   produce  a 
material    effect    on    the    scale    of   velocities    is    apparent    from    the    following    con- 
siderations. 


IN  OPEN   CANALS   OF   UNIFORM   RECTANGULAR   SECTION.  159 

Let  us  suppose  the  surface  of  the  water  to  move,  relatively  to  the  air,  with 
the  same  velocity  as  the  water  at  the  bottom  moves  relatively  to  the  bed ;  also, 
that  the  inequalities  of  the  surface  of  the  water  caused  by  the  action  of  the  air 
and  those  in  the  bed  of  the  stream  are  alike ;  and  suppose,  also,  that  a  sheet 
of  water  of  uniform  thickness,  in  contact  with  the  bed,  is  at  rest ;  we  shall  then 
have  the  water  near  the  bottom  moving  over  a  bed  of  water,  and  the  water  at 
the  surface  moving  under  a  bed  of  air,  and  as  both  beds  have  the  same  ine- 
qualities, they  will  cause  the  same  retardation  in  the  velocity  of  the  water,  except 
as  these  beds,  from  the  nature  of  the  substances  of  which  they  are  composed, 
offer  more  or  less  resistance.  These  resistances  will  be  of  the  same  nature  as  is 
experienced  by  a  body  moving  in  a  resisting  medium.  According  to  well-known 
principles,  the  retardation  in  this  case  is  as  the  square  of  the  velocity  of  the 
moving  body,  relatively  to  that  of  the  medium,  *  and  as  the  density  of  the 
medium.  The  density  of  the  air  is  about  ¥|^  of  that  of  water;  a  body  moving 
through  the  air,  with  the  same  velocity,  will  therefore  be  retarded  -g^  as  much 
as  if  it  moved  through  water.  Consequently,  in  the  case  supposed,  if  the  relative 
velocity  of  the  air  and  the  surface  of  the  water  is  the  same  as  that  of  the  bed 
and  bottom  of  the  stream,  the  retardation  at  the  surface  will  be  -gfa  of  that  at 
the  bottom.  The  retardation  being  as  the  square  of  the  relative  velocity,  if  the 
air  is  moving  in  the  opposite  direction  to  the  motion  of  the  water,  with  a  rel- 
ative velocity  equal  to  y/  840  =  29  times  the  velocity  of  the  water  at  the  bottom, 
the  retardation  at  the  surface  and  at  the  bottom  will  be  the  same,  and  the  maxi- 
mum velocity  will  be  found  at  half  the  depth. 

This  supposed  case  is  designed  merely  to  show  the  mode  in  which  the  air 
acts  in  modifying  the  scale  of  velocities,  and  to  afford  some  idea  of  the  extent 
of  its  influence. 

187.  It  follows,  from  what  is  said  in  the  preceding  section,  that  in  all  cases, 
except  when  there  is  a  wind  blowing  in  the  direction  of  the  current,  of  equal 
or  greater  velocity  than  the  water  at  the  surface  of  the  stream,  the  air  will  re- 
tard the  surface  velocity. 

Many  attempts  have  been  made  to  determine,  experimentally,  the  scale  of 
velocities  at  different  depths.  Du  Buat,  who  experimented  in  very  small  wooden 


*  The  retardation  will  be  as  the  square  of  the  velocity,  only,  when  the  inequalities  of  the  surfaces  in  con- 
tact remain  constant.  But  the  inequalities  of  the  surface  of  the  water  will  increase  and  diminish  with  the  velocity ; 
consequently,  the  retardation  at  the  surface  of  the  water  will  be  in  a  higher  ratio  than  the  square  of  the  velocity. 
It  is,  however,  sufficient  for  the  present  purpose  to  assume  it  to  be  as  the  square  of  the  velocity.  The  relative 
thickness  of  the  beds  of  water  and  air  will  also  have  an  important  effect ;  but  it  need  not  be  considered  here. 


160  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

canals,  reports  that  he  found  the  maximum  velocity  at  the  surface.  Defontaine, 
who  experimented  on  the  Rhine,  thought,  allowance  being  made  for  the  wind,  that 
the  maximum  velocity  was  at  the  surface.  Hennocque  experimented  on  an  arm 
of  the  Rhine  near  Strasburg ;  according  to  Boileau,  he  found  the  maximum  velocity 
as  follows :  — 

In  a  calm  or  very  slight  breeze  blowing  up  stream,  at  about  one  fifth  of  the 
depth  below  the  surface. 

In   a   strong   wind  blowing   up   stream,  at   about  half  the   depth. 

In   a   strong   wind   blowing   down  stream,   at   the   surface    of  the    current. 

Baumgarten,  who  experimented  on  the  canal  from  the  Rhone  to  the  Rhine, 
reports  that  he  found  the  maximum  velocity  between  one  fifth  and  one  third  of 
the  depth  from  the  surface. 

Boileau,  who  experimented  in  small  wooden  canals,  reports  that  he  found  the 
maximum  velocity  at  one  fifth  of  the  depth  below  the  surface. 

Messrs.  Humphreys  and  Abbot,*  of  the  United  States  corps  of  Topographical 
Engineers,  in  connection  with  their  operations  for  gauging  the  flow  of  the  Mis- 
sissippi, made  an  elaborate  series  of  experiments  with  submerged  floats,  to  determine 
the  scale  of  velocities.  They  report,  that,  as  a  mean  result,  they  found  the  maxi- 
mum velocity,  when  there  was  little  or  no  wind,  at  about  three  tenths  of  the 
depth  from  the  surface. 

Messrs.  Darcy  and  Bazin,  t  in  their  extensive  series  of  experiments  on  the 
flow  of  water  in  open  channels,  made  at  the  expense  of  the  French  government, 
report  that  they  found  the  maximum  velocity  below  the  surface. 

188.  In  their  work,  previously  cited,  Humphreys  and  Abbot  give  what  they 
term  the  grand-mean  curve,  determined  from  very  numerous  observations  on  the 
Mississippi  at  Carrolton  and  Baton  Rouge,  in  Louisiana,  in  the  year  1851 ;  the 
mean  depth  of  the  river  being  82  feet,  and  the  mean  velocity  3.3814  feet  per 
second.  The  curve  thus  determined  is  a  parabola,  of  which  the  equation  is 

V=  —  0.79222  dl  -f  3.2611, 

in  which  V  =  the  velocity  in  feet  per  second  at  any  depth  dlt  above  or  below  the 
axis  of  the  curve;  du  being  taken  in  fractional  parts  of  the  whole  depth  of  the 
river,  which  is  taken  as  unity;  and  the  axis  being  0.297  of  the  whole  depth  below 
the  surface. 

*  Report  upon  the  Physics  and  Hydraulics  of  the  Mississippi  River,  by  Captain  A.  A.  Humphreys  and 
Lieutenant  H.  L.  Abbot.  Philadelphia,  1861. 

t  Recherche*  ffydrauliques  entreprises  par  M.  H.  DAHCY,  continueet  par  M.  H.  BAZIN.     Paris,  1865. 


IN  OPEN   CANALS   OF  UNIFORM   RECTANGULAR   SECTION. 

If  we  put  dw  =  the  depth,  in  feet,  above  or  below  the   axis,  and  substitute   for 
a  in  the  preceding  equation  its  value  r%,  we  shall  have 


V=  —  0.00011782  dt  +  3.2611. 

The   axis  being  0.297  X  82  —  24.354   feet  below  the   surface.     When   dM  =  0,  then 
V  •=.  3.2611  feet  per  second,  which  is  the  velocity  at  the  axis  of  the  curve  and  the 
maximum.       For  the   velocity   at   the   surface   we   have   dw  =  —  24.354    and    V=: 
3.1912.     For  the  velocity  at  the  bottom  we  have  dtll  =  57.646,  and  F—  2.8696. 

189.  In   the   experiments   of  Humphreys   and  Abbot,  the   direction  of  the  wind 
was   noted   and   its   force    estimated,   a   calm    being    called    0    and    a    hurricane    10  ; 
they   made   no   experiments,  however,   when   the    force   exceeded   4.     In   the   exper- 
iments   from   which    the    grand-mean    curve    was    determined,    the    mean    estimated 
down   force   of  the   wind   is  stated  to  have  been  0.2.     They  found  that  the  direction 
and  force   of   the   wind   produced   a  marked    effect   upon    the   position   of   the   axis, 
or,   in   other  words,   upon   the   depth   below   the  surface   at  which   the   velocity  was 
a  maximum.     Their  grand-mean   curve   indicates   that  when   the   wind  was   blowing 
down   stream   with   the  force  0.2,  the  maximum  velocity  was  about  0.3  of  the  whole 
depth  ;    and   they   state   that   they  always   found   it  below  the  surface  in  a  calm,  and 
they  infer,  from   their   elaborate   experiments,  that  even  when  the  wind  was  blowing 
down  stream   with   a  velocity  equal  to   that   of  the   current,  that   the  maximum   ve- 
locity  is   generally,   if  not   always,   below   the    surface.     It   is   difficult   to   understand 
how   this   can   be  the   case   in   a    long,   straight,  uniform    channel.      The    Mississippi, 
as  is  well   known,  is   very   crooked,   and   the   disturbing   effects   of  bends  in  a  large 
stream    are   felt   at   great   distances   down  stream;    and   probably   no   point   could   be 
found   below   the   mouth   of  the    Ohio    at   which    the  velocities   in    different   parts  of 
the    section    would   be   free   from    considerable    irregularities   from    this   cause.      What 
effect   this   may  have   on  the  scale  of  velocities  does   not  appear,  but  it  will  scarcely 
be  safe  to  infer  that  it  would  be   found   to   be   the   same   in   straight   as  in  crooked 
channels. 

190.  Humphreys   and   Abbot   give  the  following  general  formulas  for  the  curve 
representing   the   scale   of    velocities,   in    any    vertical    plane   which    is   parallel   with 
the   direction   of  the   current. 

Let   V  =  the  velocity  at  any  depth. 

v  =  the  mean  velocity  for  the  whole  stream. 
Vm  =  the  mean  velocity  in  the  vertical  plane  under  consideration. 
Va  =  the  maximum  "          "  "          «          «  « 

Vn  =  the  surface        «          «  "          «          "  " 


VD  =  the  bottom 


21 


162  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

f  —  the  number  denoting  the  force  of  the  wind ;  0  being  a  calm  or  a  wind 
blowing  at  right  angles  with"  the  current,  and  10  a  hurricane ;  the 
sign  to  be  —  when  it  blows  down  stream,  and  +  when  it  blows 
up  stream. 

d  =  the  depth  below  the  surface,  at  which  the  velocity  is   V. 
da  =  the  depth  of  the  axis  of  the  curve  below  the  surface. 
D  =  the  whole  depth. 
It  =  the  mean  radius. 
Equation  of  the  curve  representing  the  scale  of  velocities, 


v-v  — 


Depth  of  the  axis  of  the  curve  below  the  surface, 

d.  =  (0.317  +  0.06  /)  5.  (2.) 

Mean  velocity, 

FJI).  (3.) 


Formulas  (1.)  and  (2.)  are  empirical,  and  founded,  mainly,  on  the  experiments 
of  Humphreys  and  Abbot.  Formula  (3.)  is  purely  geometrical,  assuming  that  the 
scale  of  velocities  is  represented  by  a  parabola. 

191.  In   gauging   the   quantity  of  water  flowing   through  our  measuring  flumes, 
numerous   observations   are   made   of  the   velocity   of  'the  tubes  in  different   parts  of 
the  width  of  the  flume,  and  their  mean  velocity  is  computed.     The  tubes  cannot  ex- 
tend quite  to  the  bottom  of  the  channel,  and  the  layer  of  water  between  the  bottom 
of  the  tubes  and  the  bottom  of  the  channel,  which  has  usually  a  less  velocity  than 
any   other  part  of   the   section,   will   not  have   its   due   weight    in    determining  the 
velocities  of  the  tubes,  which  will  therefore,  usually,  assume  velocities  a  little  greater 
than   they  would   if  they  extended   to   the   bottom.     Also,  if  the   scale   of  velocities 
at    different    depths  is   represented   by   a   parabolic   curve,   as   is  indicated  above,  in 
consequence    of  the    pressures    on   different   parts   of  the    tube   being   as   the  squares 
of  the   relative    velocities   of    the   water   and   tube,   the   tubes   will   assume   velocities 
generally,   a   little    different  from   the   mean  velocity  of  the  water  above  the   bottom 
of   the    tubes.     It    is    also   known   that   floating   bodies   do   not  generally  have   the 
same   velocity   as   the   water  in   which   they   are   immersed. 

192.  Knowing   the   mean   velocity   in   the  whole  section,  and  assuming  that  the 
formulas   of  Humphreys   and   Abbot   apply   to   our   short   channels,   we    can   compute 
the   velocity   of  the    tubes   in   the   following   manner. 

Applying   formulas  (1.),  (2.),  and  (3.)  to   the   plane   in  the  direction  of  the   cur- 


IN   OPEN   CANALS   OF   UNIFORM   RECTANGULAR   SECTION.  103 

rent,   in  which   the   mean   velocity  is   a  mean   of  the   whole   section,   we   have 


The  equation  of  the  parabola  representing  the  scale  of  velocities  is  of  the 
form  V=A-B  (d-da)\ 

The   values   of  A   and  B  can   be   determined   by   equations  (1.),  (2.),  and  (3.). 

In  (2.)  the  values  of  /  and  R  are  given  by  observation,  hence  the  value  of 
da  is  known,  which  can  be  substituted  in  (1.),  in  which,  besides  the  co-ordinates 
V  and  d,  all  the  quantities  will  be  known  excepting  Va,  which  can  be  determined 
by  (3.),  as  follows:  — 

In  (1.)  when  d  =  D  _ 


and  when  d  =  o 

y  — -  y  

Substituting  these  values  in  (3.),  also   Vm  for  v,  and  reducing,  we  have 


In  which  all  the  quantities  in  the  second  member  are  known,  and  consequently 
the  value  of  Va.  Substituting  the  value  of  Va  in  (4.),  we  have  all  the  quantities 
known  in  the  second  member,  and  consequently  the  value  of  VD.  The  values  of 
Va  and  VD  being  known,  determine  two  points  in  the  curve,  which  is  sufficient  to 
determine  the  two  constants  A  and  B  in  its  equation. 

In  experiment  1,  table  XXII.,  we  have  R  =         ''  ^o*  =  5.5656;  and  for 

,£D.  /  4o    I    *-  /\  y  .Ooo 


a  moderate  wind  down  stream  f  is  assumed   at  —  0.5.     Substituting   these   values  in 
(2,),  we  have  da  =  1.5973  feet. 

In   experiment   1,  we   can   determine   the   mean  velocity  from  the  weir  measure- 
ment and  the  section  of  the  stream. 

=  2'6719 


We  have  also  D  —  9.533. 

Substituting  these  values  in  (6.),  we  have 

Va  —  2.8979  feet  per  second. 

The  quantities  in  the  second  members  of  (4.)  and  (5.)  are  now   all   known,  and 
their  values  being  substituted,  give 

V0  =  2.0899  and   V0  =  2.8652. 


164  A  METHOD   OF    GAUGING   THE   FLOW   OF   WATER 

In  the  equation  of  the  curve   V=  A  —  B  (d-da)  , 

when  d  =  o,  then   V—  2.8652  and  2.8652  —  A  --  B  (—  1.5973)2 
and  when  d  =  9.533,  then   V  =  2.0899,  and  2.0899  =  A  —  B  (9.533  —  1.5973)2. 
From  these  two  equations  we  find 

A  =  2.8979  and  B  =  0.01283, 

and   the   equation   of  the   curve   representing   the   scale   of   velocities   in   experiment 

1  is 

V=  2.8979  —  0.01283  (d  —  1.5973)2.  (7.) 

The  curve  of  which  (7.)  is  the  equation  is  represented  by  the  line  EXD  F, 
figure  7,  plate  XV.,  OX,  Y  O  T  being  the  axes  of  co-ordinates.  Figures  8,  9,  and 
10  represent  the  curves  deduced  in  a  similar  manner  from  experiments  7,  43,  and 

47. 

193.  O  A,  figure  7,  represents  the  velocity  of  the  tube.  As  stated  above,  this 
will  generally  vary  slightly  from  the  velocity  of  the  water,  the  tube  having  such 
a  velocity  that  the  pressure  on  its  up-stream  side  will  equal  the  pressure  on  its 
down-stream  side.  These  pressures  are  due  to  the  relative  velocities  of  the  water 
and  tube;  at  the  depths  where  -the  tube  is  moving  slower  than  the  water  there 
will  be  a  pressure  on  the  up-stream  side,  and  where  it  moves  faster  than  the 
water  there  will  be  a  pressure  on  the  down-stream  side.  In  figure  7  the  portion 
of  the  tube  B  D  will  have  a  pressure  on  the  up-stream  side,  and  the  portion  G 
D  will  have  a  pressure  on  the  down-stream  side;  the  pressure  at  any  point  will 
be  proportional  to  the  square  of  the  difference  of  the  velocities  of  the  tube  and 
the  water.  Adopting  this  principle,  and  assuming  that  the  scale  of  velocities  is 
represented  by  an  equation  of  the  form 

V=A—  Bd* 

Putting  V/  —  A  X  =  the  difference  between  the  velocity  of  the  tube  and  the 
maximum  velocity  of  the  water  ;  dt  =  the  depth  A  G  to  which  the  tube  is  im- 
mersed helow  the  axis  of  the  curve,  and  retaining  the  preceding  notation,  my 
assistant,  Mr.  Joseph  P.  Frizell,  finds  for  the  case  represented  in  figure  7,  plate  XV. 


v'  ~  (dt  ~  da}  v  +  *  B  (d  - 


In   experiment   1,  table   XXII.,  we   have  dt  =  9.482  —  1.5973  =  7.8847;   da  = 
1.5973,  and  B  =  0.01283.     Substituting  these  values  in  (8.),  and  reducing,  we  have 

V*  —  0.66766  V*  +  0.44152  Vt  =  0.10650, 
from  which  we  find  V  =  0.2655. 


IN   OPEN   CANALS   OF   UNIFORM    RECTANGULAR   SECTION.  165 

Putting  V,  =  the  velocity  of  the  tube,  we  have 

Vt=Va--Vt  =  2.8979  —  0.2655  =  2.6324  feet  per  second: 
Putting    Vmt  =   the   mean   velocity   of  the   water   for  the   depth   to   which   the 
tube  is  immersed,  and    Vbt  =  the  velocity  of  the  water  at  the  bottom  of  the  tube, 
we  have  by  (7.) 

VH  =  2.8979  —  0.01283  (9.482  —  1.5973)2  =  2.1003  feet  per  second, 
and  by  (3.), 

Vmt  =  |  Va  +  I  Vbt  +  i  |  ( K  -  Vbt)  =  2.6750  feet  per  second. 

In  experiment  1  the  tube  will,  therefore,  have  a  velocity,  less  than  the  mean 
velocity  of  the  water  for  the  whole  depth  to  which  the  tube  is  immersed,  equal  to 

2.6750  --  2.6324  =  0.0426  feet  per  second, 
which  is  about  ^\  of  the  velocity  of  the  water. 

194.  The  tube  does  not  at  once  take  the  velocity  of  the  water,  but  after 
floating  a  short  distance  the  difference  is  inappreciable,  as  will  be  seen  by  the 
following  investigation. 

The  tube  when  first  placed  in  the  water  is  supposed  to  be  perpendicular  and 
at  rest,  and  to  retain  its  perpendicularity  during  its  motion ;  the  water  striking  it 
with  the  full  velocity  of  the  current  creates  a  pressure  on  its  up-stream  side ; 
yielding  to  the  pressure,  it  gradually  assumes  the  velocity  of  the  current.  It  will, 
however,  be  simpler  to  consider  the  converse  proposition,  which  will  lead  to  the 
same  result,  namely,  to  assume  that  the  water  is  at  rest  and  that  the  tube  is  im- 
pelled against  it  with  a  velocity  equal  to  the  velocity  of  the  current;  it  can  then 
be  treated  as  a  body  moving  in  a  resisting  medium  of  large  extent  in  propor- 
tion to  the  size  of  the  body. 

Let  V  =  the  initial  velocity  of  the  tube. 

v  =  the  velocity  of  the  tube  after  traversing  any  distance,  s,  in  the  fluid. 
k  =  a   coefficient,   depending   on   the   form   of  the   body,  but  which  for  a 
cylinder  moving  with  its  axis  at  right  angles  to  the  direction  of  the 
motion  is  about  0.77.      (See   Rankings  Applied  Mechanics,  London 
and  Glasgow,  1858.) 
w  =  the  weight  of  the  body. 

A  =  the  area  of  its  greatest  transverse  section  opposed  to  the  motion. 
JV=  the  specific  gravity  of  the  body. 
n  =  the  specific  gravity  of  the  fluid. 
f=  the  retarding  force. 
According   to   well-known  principles,  the   resistance  of  the   water  to   the   motion 

V2 

of  the  tube  is  k  A  n   -, 


166  A   METHOD   OF    GAUGING   THE   FLOW   OF   WATER 

i                                                                             _f        k  A  n  fa  ,-,  , 

and                                                            /  —  -       -  -  (1.) 

•>            2  g  w  V    ' 

If  the  body  is  a  cylinder,  put  D  for  its  diameter;   then  for  one  foot  in  length 

of  the  cylinder  we  have  A  =  JD,  we   have   also  w  =  £  n  IP  N.  Substituting  these 
values  in  (1.)  and  reducing,  we  have 


'~  ngDN 

In  the  case  of  a  floating  body  we  have  N  =  n,  and  consequently 


Then  (see  Button's  Mathematics,  "On  the  Motion   of  Fluids"),  giving  dv   the  nega- 
tive sign,  because  v  diminishes  as  s  increases, 


and  hence =  — ^  d  s,  (2.) 

v  n  D 

which,  by  integration,  gives 

? '  (»0 


Equation  (2.)  may  be  put  under  the  form 


dt 

*\ 

Multiplying  both  sides  by  -7-  and  reducing,  we  have 

CL  S 

d*s    -,  .          2k    j. 
—  -^(11=  —  =^  d  t 
as*  n  D 

Integratuig,  remembering  that  -y-  or  -  is  equal  to  -p>  when  t  =  o,  we  have 


Returning  to  the  real  case  and  denoting  by  s'  the  distance  traversed  by  the 
tube  in  the  time  t,  V  being  the  velocity  of  the  current,  and  vt  that  of  the  tube 
at  the  expiration  of  the  time  t,  we  shall  have 

s'  =  Vt  —  s. 

Substituting  the  values  of  s  and  t  by  (3.)  and  (4.)  and  also  V  —  ^  for  v, 
we  have 


i          n 

8  = 


IN  OPEN  CANALS  OF  UNIFORM  RECTANGULAR  SECTION.  167 

195.  By  equation  (5.)  we  see  that,  theoretically,  the  tube  never  quite  attains 
the  velocity  of  the  current ;  and  that  the  distance  it  must  float  in  order  to  at- 
tain the  velocity  of  the  current,  within  a  given  fractional  part,  is  proportional  to 
the  diameter  of  the  tube  and  is  independent  of  the  velocity  of  the  current. 

In  the  following  experiments,  s'  was  about  20  feet,  and  D  =.  2  inches  =:  £  foot. 
Substituting  these  values  in  (5.)  we  find 

V  —  v,  _  _  1 


V         ~  64 


nearly, 


That  is  to  say  :  —  a  tube  2  inches  in  diameter,  after  floating  20  feet  from  the 
point  where  it  is  put  into  the  current,  acquires  a  velocity  equal  to  about  ff 
that  of  the  current. 

196.  Observation  teaches  us  that  floating  bodies  move  faster  than  the  stream 
in  which  they  are  floating;  this  is  undoubtedly  the  reason  why  vessels  moving 
with  the  current  in  a  calm  can  be  steered  ;  they  not  only  partake  of  the  motion 
of  the  water,  but  they  have  an  independent  motion  due  to  the  inclination  of  the 
surface  of  the  water;  the  constant  intermingling  of  the  upper  and  lower  parts  of 
a  stream  prevents  the  water  at  and  near  the  surface  from  attaining  a  velocity 
as  great  as  it  otherwise  would.  Navier*  has  investigated  this  subject;  assuming 
that  the  velocity  of  the  water  is  uniform  to  the  depth  to  which  the  body  is 
immersed,  he  finds,  adopting  our  own  notation, 


in  which 

V,  =  the  excess  of  the  velocity  of  the  floating  body  over  that  of  the  water. 
g  =  the  velocity  imparted  by  gravity  in  one  second. 
Q  =  the  volume  of  water  displaced  by  the  floating  body. 
/  =  the  slope  of  the  surface. 

k  =  a  coefficient  depending  on  the  form  of  the  body. 
A  =  the  area  of  the  greatest  transverse  section  of  the  body. 
In   these   experiments   the   floating   bodies   are    cylinders   with    the    axes  vertical, 
for   which    case    k   is   nearly   0.77   (art.    194).      Put   L   for    the    length    of    the    im- 
mersed part  of  such  a  cylinder  and  D  for  the  diameter  ;    then 

Q  =  |  7i  I?  L,  and   A  =  D  L. 

Substituting   these  values,  and   also   the  values   of  g  and   k,  in   the   above  equation, 
and   reducing,  we   have 

Ve  =  8.1  ^D~I.  (7.) 

*  Architecture  Hydraulique,  par  BELIDOR.     Paris,  1819,  page  358. 


1G8 


A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 


The   value    of  /  can   be    determined   from    Eytelwein's    formula   for   the    motion 
of  water   in   open   channels,   which   when    the    English   foot   is   the    unit   is  * 


R  1=  0.000  024  265  1  v  -f  0.000  111  415  5  v2. 


(8.) 


In  which  R  is  the  mean  radius,  /  the  descent  in  the  unit  of  length,  and  v  the 
mean  velocity. 

Formula  (7.)  indicates  that  the  excess  of  velocity  is  proportional  to  the  square 
root  of  the  diameter  of  the  tube,  and  also  to  the  square  root  of  the  slope.  Except 
in  very  small  velocities,  the  velocity  of  the  current  is  nearly  proportional  to  the 
square  root  of  the  slope ;  consequently,  the  excess  of  the  velocity  of  the  floating 
body  over  that  of  the  fluid  in  which  it  is  floating  is  nearly  proportional  to-  the 
velocity  of  the  current,  except  when  the  latter  velocity  is  very  small. 

In  experiment  1  we  have  R  =  5.5656  and  v  =  2.6719  (art.  192) ;  substituting 
these  values  in  (8.)  we  find  1=  0.000  154  56,  we  have  also  Z>=£;  substituting 
these  values  in  (7.)  we  find  Vt  =  0.0411  feet  per  second,  which  is  about  g^  of 
the  mean  velocity  of  the  water.  Neglecting  the  small  effect  this  excess  of  velocity 
would  have  on  the  velocity  deduced  from  the  equality  of  the  pressures  on  the  up- 
stream and  down-stream  sides  of  the  tube,  we  find  for  the  computed  velocity  of 
the  tube  in  experiment  1,  2.6750  —  0.0426  +  0.0411  =  2.6735  feet  per  second; 
which  differs  0.0015  feet  per  second,  or  TTVg>  from  the  mean  velocity  of  the  water 
for  a  depth  equal  to  the  length  of  the  immersed  part  of  the  tube,  determined 
by  the  formulas  of  Humphreys  and  Abbot.  The  mean  velocity  of  the  tube  by 
experiment  was  2.6830  feet  per  second,  which  exceeds  the  computed  velocity  by 
0.0095  feet  per  second,  or  ^|T.  Similar  computations  have  been  made  for  exper- 
iments 7,  43,  and  47,  table  XXII.,  which  are  selected  as  giving  a  wide  range  of 
conditions.  The  data  and  results  are  given  in  the  following  table. 

TABLE     XIX. 


1 

a 

3 

4 

5 

6 

7 

8 

9 

1O 

11 

Depth  of  the 

Parameter  of 

No. 
of  the 

Depth  of 
water  in 

Mean  Radius. 

Length 
of  the 
immersed 

Mean  velocity 
of  the  water 
deduced  from 

Assumed 
value  of 

axis  of  the 

parabola, 
representing 
the  scale  of 

the  parabola, 

representing 

Maximum 
velocity  of 

Velocity  of 
the  water  at 

Velocity  of 
the  water  at 

!>.]'. 

the  flume. 

part  of 

the  weir 

/ 

velocities, 
below  the 

velocities. 

the  water. 

the  surface. 

the  bottom. 

the  tube. 

measurement. 

surface  of 

the  water. 

D 

R 

d. 

V 

d. 

B 

Va«A 

V0 

VD 

Feet. 

feet. 

Feet  per 
Second. 

Feet. 

Feet  per 
Second. 

Feet  per 
Second. 

Feet  per 
Second. 

1 

9.533 

5.5656 

9.482 

2.6719 

—  0.5 

1.5973 

0.01283 

2.8979 

2.8652 

2.0899 

7 

9.530 

5.5645 

8.530 

2.6539 

—  0.1 

1.7306 

0.01280 

2.8686 

2.8302 

2.0902 

43 

8.172 

5.07-23 

7.120 

0.4961 

0.0 

1.6079 

0.00777 

0.5871 

0.5670 

0.2521 

47 

8.165 

5.0696 

8.122 

0.4842 

—  0.3 

1.5158 

0.00770 

0.5777 

0.5600 

0.2374 

A  Treatise  on  Water- Works,  by  CHARLES  S.  STORROW.     Boston,  1835. 


IN   OPEN   CANALS   OF   UNIFORM    RECTANGULAR   SECTION. 


109 


TABLE     XIX.  —  CONTINUED. 


13 

13 

14 

15 

16 

17 

18 

19 

No. 
of  the 
Exp. 

Mean  velocity 
of  the  water 
for  a  depth 
equal  to  the 
length  of  the 
immersed  part 

Velocity  of  the  tube 
dedured  from  the 
formula  founded  on 
the  equality  of  the 
pressures  on  the 
up-stream  and 
down-stream  sides 

Difference 
between  the 
velocities  in 
column  12 
and 
column  13. 

Slope  of  the  surface 
of  the  water  in  the 
flume,  deduced  from 
Eytelwein's  formula 
for  the  motion  of 
water  in  open 
channels. 

Excess  of  the 
velocity  of 
the  tube  over 
that  of  the 
water  in 
which  it  is 
fl  oatinp, 

Computed 
Telocity  of 
the  tube. 

Mean 
velocity  of 
the  tube  by 
experiment. 

Difference  between  the  velocity  of 
the  tube  by  computation  and  by 
experiment. 

o        cue. 

of  the  tube. 

deduced  from 

Absolute 

Proportional 

Navier's 

difference. 

difference. 

formula. 

'  mt 

vt 

Feet  per 

/ 

V. 

Feet  per  Sec. 

Feet  per  Second. 

Second. 

Feet  per  Sec. 

Feet  per  Sec. 

Feet  per  Sec. 

1 

2.6750 

2.6324 

0.0426 

0.000  154  56 

0.0411 

2.6735 

2.6830 

+  0.0095 

+  0.0036 

7 

2.7088 

2.6752 

0.0336 

0.000  152  60 

0.0409 

2.7161 

2.7260 

-j-  0.0099 

+  0.0036 

43 

0.5247 

0-5108 

0.0139 

0.000  007  78 

0.0092 

0.5200 

0.5190 

—  0.0010 

—  0.0019 

47 

0.5111 

0.4669 

0.0442 

0.000  007  47 

0.0090 

0.4759 

0.4950 

+  0.0191 

+  0.0401 

197.  It   will   be    seen,   by    column    19   in    the    preceding   table,  that   the    differ- 
ences  between  the    computed    and    observed  velocities  are  not  very  regular ;   perhaps 
as   much   so,   however,   as   could   be    anticipated,   considering    the   wide    difference   in 
the    conditions    in    the    experiments    of    Humphreys   and   Abbot    and   in   the    exper- 
iments   at    the    Tremont    measuring    flume,    and    that    their    data    for    determining 
formulas   (1.)    and  (2.)    are    not   of    a   character    to   afford   much   confidence    in   their 
application  to  cases  where  the  conditions  are  so  different. 

198.  From    the    preceding   investigation   we   infer,  that   in   rectangular  channels, 
in   which    the    natural  .scale    of  velocities   at   different   depths   is  established,  and   the 
surface    velocity    not    very    much    retarded    by   the   wind,   the    tube    is   retarded   on 
account   of  the    pressures    on  the  tube  being  as  the  squares  of  the  relative  velocities 
of  the   water   and   tube    at   different   parts   of  its   length,   and   is   accelerated    by  the 
independent   motion  of  the   tube   due   to  the  slope  of  the  surface  of  the  water,  and 
that  the    retardations    and   accelerations   compensate    each  other   to   a  greater  or  less 
degree    under   different   circumstances. 

Taking  a  mean  of  the  four  experiments  in  table  XIX.,  the  computed  velocity 
of  the  tube  is  about  Jg-  less  than  the  observed  velocity;  and  assuming  this  rela- 
tion to  be  of  general  application,  we  might,  evidently,  by  a  process  the  reverse 
of  that  by  which  table  XIX.  is  computed,  from  the  observed  velocity  of  the  tube, 
arrive  at  the  mean  velocity  of  the  water  in  the  flume.  It  would,  however,  involve 
lengthy  computations,  and  the  result  would  not  be  free  from  uncertainty,  on  account 
of  the  doubtful  applicability  of  the  formulas  of  Humphreys  and  Abbot;  and  how- 
ever interesting  such  an  investigation  might  be  as  a  scientific  matter,  it  will  be 
safer,  in  practice,  to  rely  upon  rules  deduced  from  suitable  experiments,  even  if 
such  rules  are  empirical. 

199.  In   arranging   the   programme   of   these   experiments,   it   was    designed   to 
make   them   under   the   various   circumstances    which   occur   in   the   gaugings   in   the 

22 


170  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

several  measuring  flumes  at  Lowell,  and  as  nearly  as  practicable  on  the  same  scale ; 
the  only  material  deviation  from  what  was  desired  in  the  latter  respect,  was  in 
the  width  of  the  channel;  this  was  necessarily  limited  to  the  width  of  the  canal 
in  which  the  experimental  flume  was  placed.  A  series  of  experiments  with  tubes 
of  seven  different  lengths,  and  with  velocities  varying  from  2.7  to  0.5  feet  per 
second,  was  made  with  a  flume  of  as  great  a  width  (26.745  feet)  as  could  con- 
veniently be  made  in  the  canal,  and  another  series,  similar  in  respect  to  length 
of  tubes,  but  with  velocities  varying  from  5.0  to  1.4  feet  per  second,  was  made 
with  a  flume  of  half  the  width  of  the  preceding. 

200.  The    experiments   consisted  in  making   a   gauge  of  the  quantity  of  water 
passing    the    measuring    flume,    by   observing    the   velocity   of   loaded    tubes   floating 
down   different   parts   of  the    section   of  the   flume,  and   from   these   observations  de- 
ducing  the   mean   velocity  of  the    tubes   for   the   whole   section ;   this  mean  velocity 
is   provisionally   assumed   to   be    the    mean   velocity   of  the   water  in   the   flume,  and 
when   multiplied   into   the   area  of  the   section   gives   the    quantity  of  water   passing 
the   flume    according   to   the  flume  measurement.     After   leaving   the   flume,  the  same 
volume   of  water  is   made   to   pass   over  a  weir,   and   the   depth  on  the  crest  being 
observed,   the    quantity   is   computed   by   means   of    a    formula   determined   from   the 
experiments   made   at   Lowell,  in    1852,   and   previously  described  in  this  work.     The 
quantity  thus   computed  (with   a   minute    correction   for  leakage   in   the   experiments 
on   the   narrow   flume)   is   taken   as   the    true    quantity   passing   through  the  measur- 
ing  flume,   and   the    comparison   of   this   quantity   with   that   obtained   by   the   flume 
measurement   determines  the   correction   in   that   particular   experiment. 

201.  Figures    1  and    2,   plate    XVI.,  are    a  general  plan  and  longitudinal  section 
of  the    entire    apparatus   used    in   the    experiments   with   the  wide    flume.     A.   is   the 
Northern    Canal,    through   which    the    principal    supply   of   water    is    primarily   con- 
ducted  from    the    Merrimack    River    to    the   manufacturing   establishments.     JB,   the 
Tremont   Gates,   through  which   water  is  at  times  drawn,  to  make  up  any  deficiency 
in    the    supply   in    the    lower    level    of    the   Western    Canal.      C  is   a   grating   put 
across  the  canal  for  the  purpose  of  equalizing  the  flow  of  the  water  in  different  parts 
of  the    section   of  the   canal.      D   is   a   raft   or   float   for   the    purpose   of    destroying 
the   oscillations   of  the  surface,  caused   by    the    admission   of  the  water   at   the  gates 
B,  and   which   oscillations   were   partially  propagated    through   the    grating  C.     With- 
out  the    float   the   oscillations    of    the    surface    extended    into    the    measuring    flume, 
and   imparted    corresponding   vertical   oscillations   to    the    tubes,  causing  those  extend- 
ing  nearly   to    the    bottom    to   touch   occasionally,   which   would    of    course   tend   to 
retard   them.     E  is   the   measuring    flume.      F  the   Tremont    Wasteway,   over  which 
the   occasional   supply  from   the   Tremont   Gates   passes   into   the  lower  level   of  the 


IN  OPEN   CANALS  OF   UNIFORM  RECTANGULAR  SECTION. 

Western  Canal,  W;  on  this  wasteway  is  erected  the  weir  for  gauging  the  water 
after  it  has  passed  through  the  measuring  flume. 

Figures  1  and  2,  plate  XV.,  are  a  plan  and  transverse  section  of  the  wide 
flume.  The  original  section  of  the  canal  is  lined,  from  A  to  B,  with  planks  about 
2.25  inches  in  thickness,  planed  on  the  surface  in  contact  with  the  current,  and 
fastened  to  timbers  which  are  securely  bolted  to  the  side  walls  and  to  stones 
sunk  in  the  bottom  of  the  canal  for  the  purpose.  The  lining  plank  is  connected 
with  an  old  piling,  C  D,  put  in  for  another  purpose,  which  extends  through  the 
side  walls  of  the  canal  and  into  the  earth  on  each  side,  effectually  preventing 
any  flow  of  water  outside  of  the  plank  lining.  E  F  represents  an  obstruction  in 
the  canal,  used  in  a  portion  of  the  experiments  for  the  purpose  of  creating  irreg- 
ularities in  the  flow  through  the  measuring  flume.  G  is  a  float  of  timber  and 
plank  for  the  purpose  of  destroying  the  oscillations  of  the  surface  of  the  water 
caused  by  the  obstruction  E  F.  The  obstruction  and  float  were  used  only  in  exper- 
iments 123  to  140,  which  do  not  form  any  part  of  the  series  from  which  the 
formula  of  correction  is  deduced. 

202.  Figures  3  and  4,  plate  XV.,  represent  the  same  measuring  flume  as 
figures  1  and  2,  with  the  changes  made  for  the  purpose  of  narrowing  the  flume. 
The  partition  A  B  was  placed  near  the  middle  of  the  flume ;  the  dam  C  pre- 
vented any  flow  of  water  through  the  part  of  the  flume  shut  off  by  the  partition. 
In  order  to  make  the  flow  through  the  narrow  flume  more  nearly  like  that  through 
a  long  canal  of  uniform  section,  and  in  this  respect,  more  like  the  flow  through 
the  wide  flume,  the  partition  was  extended  above  the  flume  from  A  to  D,  a 
distance  of  about  100  feet.  This  extension  of  the  partition  was  constructed  of 
planks,  the  lower  ends  of  which  were  set  in  the  earth  forming  the  bottom  of 
the  canal,  and  the  upper  ends  were  secured  to  timbers  and  stayed  as  represented 
in  figure  3.  The  part  of  the  partition  from  A  to  D  was  intended  to  be  as 
nearly  impervious  to  the  passage  of  water  through  it  as  it  could  be  conveniently 
made  without  jointing  the  planks ;  the  partition  from  A  to  B  was  made  with  more 
care  and  was  intended  to  be  water  tight;  the  lining  of  the  flume  was  also  in- 
tended to  be  water  tight;  neither  lining  nor  partition  were,  however,  quite  tight. 
In  the  experiments  with  the  wide  flume,  no  difficulty  was  experienced  from  this 
cause ;  in  the  experiments  with  the  narrow  flume  it  was  necessary  to  ascertain 
the  correction  to  be  applied  on  account  of  the  leakage.  It  would  occupy  much 
space  to  give  an  intelligible  description  ,of  the  operations  performed  to  arrive  at 
the  correction  to  be  made  on  this  account,  and  as  it  was  found  to  be  very  small, 
less  than  -j^V^  part  of  the  quantity  passing  the  flume  in  any  experiment,  further 
mention  of  it  is  unnecessary. 


172  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

203.  The   whole   length   of  the   measuring  flume   was   about   100  feet,  only  70 
feet,  however,   was   included   between   the    upper    and   lower   transit   stations  H  and 
I ;   the    principal   part   of  the   remainder,  A  H,   being   about  28.5   feet,   was   used  as 
an  entrance  or  mouth-piece    to   the   part  used  for  ascertaining  the  velocity,  in  order 
that   the    eddies   and   other   irregularities   incident   to   the    small  change  in   the   form 
and   dimensions   of    the   canal,   might   be,   to   some   extent,   obliterated,   before   reach- 
ing  the   part  of  the   flume  used  for  ascertaining  the  velocity.     This  space  was  also 
serviceable   by  giving  opportunity  for   the   tubes   to   become   free   from   considerable 
oscillations   and   to   attain,   sensibly,   the   velocity   of  the    current. 

204.  Figures  5  and  6,   plate  XV.,  represent  two  of  the  loaded  tubes,  used  for 
ascertaining  the  velocity  of  the  water  in   the   flume.     Figure  5  represents  the  tube 
used    in    experiment    1,   in   which   it   extended    as    nearly   to   the   bottom,  E  E,   as 
appeared   to   be   safe    and   not    touch    during    its   passage.      Figure    6    represents   the 
tube   used   in   experiment   7,  in   which   the   space   between  the   bottom   of  the   tube 
and   the   bottom   of  the   canal  was   about   one   foot.      The   tubes   are   cylinders,   two 
inches   in    diameter,   made   of   tinned   plates,    soldered  together,  with  a  piece  of  lead, 
C  B,  of   the   same   diameter,   soldered   to   the   lower   end,   and   of   sufficient  weight 
to   sink   the    tube   nearly   to   the   required   depth,  which   was  such  as  to  leave  about 
four   inches   of  its   length    above   the    surface   of  the   water.     The  required  depth  of 
immersion   was   marked   with   red   paint   at   A.     In   order   to    adjust   it  precisely,  the 
tube   was   placed   in   a  tank   made   for   the   purpose,   and  small   pieces   of  lead   were 
dropped   into    the    top    of   the   tube ;    these   rested   on   the   mass   of  lead,   C  B,   and 
were   added   until   the    tube    was   sunk   to   the    required   depth ;    the    orifice    D    was 
then   closed   with   a  cork.     The   tubes   were   allowed   to   remain   floating  in  the  tank 
for   some   time   after   they   were   adjusted,  in   order  to  ascertain  whether  they  leaked 
or  not ;    if  they   did   they   were   taken   out  of   the   tank   and   filled   with   water,  in 
order  to   ascertain   the   position  of   the   leak,   which  was   then   stopped  with    solder 
and    the    operation   of    adjustment   repeated.      The    centres   of    gravity   of    the    tubes 
thus   adjusted   were  at    G,   G  B  in  figure  5   being   about   1.90  feet,  and  in  figure  6 
about    1.78    feet.     The    centres   of   gravity   being    so   low,  the   tubes   had   a   strong 
tendency   to   maintain   a  vertical   position.     The    velocity  of  the    current  being,  how- 
ever,  generally   more   rapid   near  the  surface  than   near  the  bottom,  the  upper  parts 
of  the   tubes   must,  of  course,   generally,   have   had   an   inclination  down  stream;   no 
special   observations   were    made    of    the    amount    of   inclination ;    in    the   small   part 
projecting    above    the    surface    of    the    water   none    was    apparent,    and    as    it    was 
evidently  very   small,  it   has   been   assumed   in   all   these    experiments  that  the  tubes 
constantly   maintained    a   vertical   position. 

Tubes   of  thirty-three   different  lengths,  from   six   feet  to   ten   feet,  six   of  each 


IN   OPEN    CANALS   OF   UNIFORM   RECTANGULAR   SECTION.  173 

length,  had  been  previously  provided  for  the  ordinary  measurements  of  the  water 
used  by  the  manufacturing  companies.  From  this  stock  three  or  four  of  each 
length  required  for  these  experiments  were  selected  and  specially  adjusted  for 
each  experiment. 

The  tubes  were  put  into  the  water  by  an  assistant  standing  upon  the  bridge 
K,  figure  1.  plate  XV. ;  it  is  done  by  a  manoeuvre  requiring  a  little  practice  to 
perform  it  satisfactorily.  The  assistant  stands  with  his  face  up  stream,  with  the 
tube  in  hand,  the  loaded  end  directed  downwards,  but  up  stream,  at  an  angle 
with  the  horizon,  greater  or  less,  depending  on  the  velocity  of  the  current.  At 
a  signal,  he  pushes  the  tube  rapidly  into  the  water  at  the  angle  at  which  he 
previously  held  it,  until  the  painted  mark  near  the  upper  end  of  the  tube  reaches 
the  surface  of  the  water,  he  retains  his  hold  of  the  upper  end  of  the  tube  until 
the  current  has  brought  it  to  a  vertical  position,  when  he  abandons  it  to  the 
current;  he  then  turns  round  and  observes,  at  its  passage  under  the  transit  timber 
H,  how  far  the  tube  is  from  the  left  side  of  the  flume,  the  up-stream  face  of 
the  timber  being,  for  this  purpose,  graduated  in  feet,  and  distinctly  marked 
and  numbered.  He  also  observes  its  passage  under  the  middle  timber  L,  and  the 
lower  transit  timber  /  in  a  similar  manner.  As  he  makes  the  observations  he 
calls  the  distances,  which  are  recorded  by  another  assistant.  The  mean  obtained 
by  adding  together  the  observed  distances  at  the  upper  and  lower  transit  timbers, 
and  twice  the  observed  distance  at  the  middle  timber,  and  dividing  the  sum  by 
four,  is  taken  as  the  mean  distance  of  the  tube  from  the  left  side  of  the  flume 
during  its  passage. 

205.  The  up-stream  sides  of  the  timbers  H  and  /  are  vertical,  and  70  feet 
apart,  and  form  the  upper  and  lower  transit  stations.  The  times  when  the  tube 
passes  the  transit  stations  are  noted  by  an  observer  at  .2V,  who  has  a  marine 
chronometer  on  a  table  before  him.  The  passage  of  the  tube  at  the  transit 
stations  is  observed  by  assistants  who  are  seated  at  M  and  0.  The  signals  of 
the  transits  are  communicated  to  the  observer  of  the  times  by  means  of  an  electric 
telegraph  erected  for  the  purpose ;  connected  with  the  telegraph  are  two  break- 
circuit  keys  which  are  conveniently  placed  within  reach  of  the  assistants  at  M 
and  0,  and  a  telegraphic  call  is  placed  on  the  table  at  .2V,  near  the  chronometer. 
When  the  tube  has  been  abandoned  to  the  current  by  the  assistant  on  the  bridge 
K,  the  assistant  at  M  puts  one  of  his  eyes  in  the  vertical  plane  forming  the 
upper  transit  station,  and  at  the  instant  when  the  tube  passes  this  plane  he  de- 
presses the  key  of  the  break-circuit,  which  causes  a  signal  to  be  made  at  the  call 
near  the  chronometer,  the  observer  at  N  noting  the  time  when  the  signal  is 
made.  The  chronometer  marks  half  seconds  only,  but  the  times  are  noted,  by 


174  A   METHOD   OF    GAUGING   THE   FLOW  OF   WATER 

estimation,  to  tenths  of  seconds.  (Art.  142.)  The  difference  of  the  observed  times 
of  the  transits  at  the  two  stations  gives  the  time  during  which  the  tube  passes 
the  70  feet;  dividing  the  distance  by  the  time,  the  quotient  is  the  velocity  in 
feet  per  second.  Another  assistant  observed  the  depth  of  the  water  in  the  flume ; 
this  was  done  during  the  passage  of  each  tube ;  the  height  of  the  water  was 
observed  in  the  box  P,  figure  1,  plate  XV.,  placed  between  the  lining  planks  and 
the  wall  of  the  canal ;  there  was  a  communication  between  this  box  and  the  flume 
by  means  of  a  pipe,  which  opened  into  the  flume  near  the  timber  L,  and  about 
four  feet  above  the  bottom  of  the  flume.  The  box  P  contained  a  scale  graduated 
to  hundredths  of  feet,  the  zero  point  of  which  was  at  the  mean  elevation  of  the 
bottom  of  the  part  of  the  flume  between  the  transit  stations  H  and  /.  The 
bottom  of  the  flume  was  very  nearly  horizontal,  the  elevations  to  obtain  the  mean 
were  taken  at  32  points,  the  extreme  difference  observed  was  0.027  feet. 

206.  Printed   forms,   bound   up   in   books,   were    prepared,   in    which    the    obser- 
vations  were  entered.     Table  XX.  compiled  from  three  of  these  books,  contains  the 
observations   made   in   experiment   No.  1,  together  with   some   of  the   steps   towards 
obtaining  the   quantity   of  water.     The   distances   given   in   column   1  were  arranged 
and   entered   previous   to   commencing    the    experiment,  and  were  called  in  order,  for 
the  information  of  the  assistant  who  put  in  the  tubes,  by  the  assistant  who  observed 
the    times   of    the   transits,   as   he    became    ready   to    make    the    observations.      The 
intervals  of  time,  given   in   column   4,  are   the   differences  of  the   times  of  the  tran- 
sits  given   in   column   3.     The  velocities   of  the   tubes  given  in  column  5,  are  taken 
from    table   XXVIII.,   which   has   been   computed,   for   the   purpose  of  facilitating  the 
ordinary    measurements    of    the    water   used    by    the    manufacturing    companies    at 
Lowell. 

207.  To   find   the   mean  velocity   of  the   tubes,   all  the   observed  velocities  are 
plotted   on   section   paper,   engraved   for   the   purpose;  reduced   copies   of  several   of 
these   diagrams   are    given   in   plate   XVII.     The    ordinates   of  the  irregularly  curved 
line   are    intended    to    represent    the    mean    velocities    of    the   tubes    at    the   corre- 
sponding   points   in   the   width    of    the   flume ;    this    line    is   drawn   on   the    original 
diagram   by   the    eye,   which   it   is   plain   cannot  lead   us   much   astray.     The  area  of 
the   figure   A   B   C  D,   experiment    1,    divided    by    the    width    of   the    flume,  will 
evidently   give   the    mean    velocity   of    the    tubes.      The    areas   in   experiment    1  for 
each   foot    in   width,    excepting    the   last,   are    given   in   column    A,   table   XX.;   the 
sum   of   these   areas    is    71.768,   which   being   divided   by    26.746,   the   width   of  the 
flume,   gives   2.6833   feet   per  second   for   the  mean  velocity  of  the  tubes.     This  last 
quantity,    (assuming   it   to   be   the   same   as   the   mean    velocity   of  the   water,)   mul- 
tiplied by  the  area  of  the  transverse  section  of  the  stream,  which  in  this  experiment 


IN   OPEN   CANALS   OF   UNIFORM  RECTANGULAR   SECTION.  175 

is  26.746   X   9.533  =  254.97  square  feet,  gives  684.16  cubic  feet  per  second,  as  the 
quantity  of  water  passing,  according  to  the  flume  measurement. 

208.  It  will   be   perceived,   by  reference    to   the    diagrams   in   plate    XVII.,  that 
the  observed   velocity   at   the  same   part   of  the   section   is   constantly   varying;    this 
is   not   due,   in   any   sensible    degree,  to   errors   of  observation,  but  to  actual  changes 
in  the  velocity,  due  to  the    unstable   condition   of    the   current.      In   all   these  exper- 
iments,  the   area  of  the    section,  and   the    quantity  of    water   flowing,   were    sensibly 
constant   throughout   an   experiment;    the    mean    velocity    must,    consequently,    have 
been   nearly   constant,   and   the   only   explanation   of  the   observed   variations   in   the 
velocity  is,  that   there  was  a    constant   interchange   of  place   of  currents   of  different 
velocities. 

209.  The   water  after  leaving   the   measuring  flume   passed  to  the  weir  erected 
on   the    Tremont   Wasteway,   F,   figures  1  and  2,  plate  XVI.     This  weir  was   in  •  two 
divisions,   each   having   about   40    feet   in   length   of  water-way;    the    Westerly   divis- 
ion,  and    a    part    of   the    Easterly   division,   are    represented    on    an  enlarged   scale 
by  figures   3  and  4.      Figure    5   is   a  sectional   elevation   of  the   weir   and   some   of 
the    apparatus    connected   therewith.     A   is   a   grating   for   the   purpose   of  equalizing 
the    flow   towards   the   weir,   and    for   obliterating    the    irregularities   in   the   direction 
of  the   currents   approaching   the    weir,   which   it   is   obvious,   from    an   inspection   of 
the   form   of  the   approaches,   would   have   otherwise   existed.     The   whole   length  of 
the   grating  was   88   feet;    the    vertical   slats   were   4   inches   wide,   in   the   direction 
of  the   current,   and   one  inch   thick,  the   spaces   between   the   slats,  for   the   passage 
of    the   water,  were    about    1.125   inches    wide.      To   equalize    the    flow   still   further, 
horizontal   slats    1.5    inches   wide   were  placed  on  the    up-stream   side  of  the  grating; 
they   were    placed   principally   at   the   Westerly   part    of    the  grating,   on   which    the 
current    from   the   measuring   flume   impinged   most   directly.      The   whole   length   of 
the    grating    being    divided    into    five    nearly    equal    parts,    the   Westerly   part   had 
eight   horizontal    slats,   the    next   part  had    six   slats,    the   next   four,   the   next   two, 
and   the    next,   or   most   Easterly   part,   had   none.      The    effect    of    this   grating   was 
to    obliterate    all    sensible    lateral    currents ;    it   did    not,    however,    entirely    equalize 
the    flow,  except  in  a  small  portion  of  the  experiments.     In  experiment  1,  in  which 
the    discharge    over   the    weirs   was    681.25   cubic   feet    per   second,   the    mean   depth 
on    the    Easterly   division   of    the    weir   was    0.0387   feet   less   than   on   the   Westerly 
division ;   in  experiments  43  to  49,  in  which  the    mean   discharge   was    106.05   cubic 
feet  per  second,  the   mean   depth  on  the   Easterly  division   of  the  weir  was  0.00026 
feet  greater  than   on  the  Westerly  division.      In   computing   the  discharge  the  mean 
of    the    observed    depths    on   the    two    divisions   of  the   weir  is    taken,   the   small   in- 
equalities in   the  depths   on    the    two   divisions   produce   inappreciable    effects  on   the 
results. 


176 


A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 


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IN   OPEN   CANALS   OF   UNIFORM   RECTANGULAR  SECTION.  177 

210.  The   up-stream  face  of  the   weir  F  P,  figure   5,   was   a   vertical   plane,   6 
feet  in  height  and  88  feet  long;    the  crest  of  the  weir  was  of  the  form  represented 
by  figure  3,  plate  XVIII.,  and  was  horizontal  for  a  width  of  0.5  inches ;    the  up-stream 
edge  presented  to  the  current  was  as  sharp  as  could  be  conveniently  maintained  in 
wood ;    the  down-stream  side  of  the  cr*est  was  chamfered  off  at  an  angle  of  45°  with 
the  vertical.     The   two  divisions  of  the  weir  were  separated  by  a  space   B  four  feet 
wide,  and  at  each  of  the  ends  C  there  was  a  space  of  two  feet;   the  up-stream  faces 
of  these    spaces  were   in   the  same  vertical  plane  as  the   up-stream  face  of  the  weir, 
and    were    deemed   to   be    ample    to   insure    complete    contraction  at  the  ends  of  the 
sheets  of  water.      The   dam   or  wasteway  on  which    the  weir  was    erected  was   of  a 
form  adapted  to  the  convenient  discharge   of  water  over  its  crest,  and  for  the  regu- 
lation   of  the    flow   over   the    same;    this   was,  however,  not   the   form   to  which   the 
ordinary  formula   for   computing    the    flow  over  a   weir  applies,  and  it  was  therefore 
necessary  to   make    such  changes   in   the  form  of  the  crest  as  would  permit  of  such 
application.      It  was  not  deemed   admissible    to   take    down   the    top  of  the   existing 
dam,  and  to  reconstruct  it   of  suitable  form ;   all   that   could    be   done  was   to   make 
additions  which  could  be  removed  when  the  experiments  were  completed. 

211.  In  order  to  preserve  a  sufficient   depth    of  flow   over   the  weir,   the    crest 
could    not   be   raised   more    than    one    foot   above    the   wasteway.     The    standards   D, 
figures  3  and  5,  which  formed  part  of  the  wasteway  and   were    required   to    support 
the  flash-boards  used  in  regulating  the  flow  over   the  wasteway,   it  was  necessary  to 
leave  undisturbed ;    in    order   that   they  should  not   obstruct  the    flow  over  the  weir, 
the  crest  of  the  latter  was  placed  at  a  certain  distance  up  stream ;    this  was  accom- 
plished  by   fastening   the   large    timber   E,   figure    5,   to    the    up-stream   face    of    the 
wasteway,  the  plank  F,  figures  3  and   5,  forming  the  crest  of  the  weir,  was  fastened 
to   this    timber.      As   thus   arranged,   the    sheet   of  water   passing   over   the   weir   fell 
vertically,  and  with  very  slight  obstructions,  to  the  cap  of  the  wasteway,  and  passed 
horizontally,  a  distance  of  about  1.4  feet  from  the  up-stream  face   of  the  weir  plank 
F,  before  it  struck  the  standards  D. 

212.  The  weir  was  made   in    two    divisions   for   the   purpose    of  facilitating   the 
passage    of    air    under    the    sheet,   former   observations   having   shown   that  air   thus 
situated  is  rapidly  carried    away   by  the   water,   and  unless  sufficient  means  are  pro- 
vided    for    renewing    it,    its    place    will     be    speedily    taken    by    water,    which    will 
materially  affect  the  flow  over  the  weir  and  prevent  the  correct   application   of  the 
formula   for   computing   the    discharge.      This   precaution  proved,  however,   to   be   in- 
sufficient to  prevent  the  space  under  the  sheet  from  becoming  filled  with  water;   it 
was  evident  that  a   portion  of  the    water   striking   the    top   of  the    wasteway   flowed 
back  towards  the  weir  and  filled  the  space  which  ought  to  be  kept  free;  to  prevent 

23 


178  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

this,  the  board  G,  figures  3  and  5,  was  put  on;  its  width  was  sufficient  to  reach 
from  the  top  of  the  timber  E  very  nearly  to  the  underside  of  the  sheet ;  this 
remedied  the  difficulty  in  a  great  degree,  but,  unless  the  width  of  the  board  was 
properly  adjusted  to  the  sheet,  it  failed  to  operate  satisfactorily;  if  too  low,  the 
water  flowed  back  over  the  top,  if  too  high,  the  sheet  of  water  struck  the  board, 
in  either  case  very  soon  filling  up  the  space  between  the  board  and  the  weir 
plank;  at  first  the  only  escape  of  the  water  from  the  trough  formed  by  the  board 
G  and  the  weir  plank  was  at  the  ends,  and  the  trough  being  forty  feet. long,  the 
escape  from  the  central  parts  was  very  slow.  This  difficulty,  however,  was  remedied 
by  attaching  leaden  pipes,  two  inches  in  diameter,  to  the  board  G;  these  pipes 
were  about  sixteen  feet  long  and  were  laid  on  the  inclined  surface  of  the  apron 
of  the  wasteway,  the  lower  ends  of  the  pipes  being  about  five  feet  below  the  upper 
ends.  The  Easterly  division  was  first  fitted  up  with  twenty-six  of  such  pipes ;  upon 
trial  this  proved  to  be  a  much  greater  number  than  was  necessary  to  afford  escape 
for  the  water  flowing  back  over  the  top  of  the  board  G,  and  the  Westerly  division, 
which  is  that  shown  on  figure  3,  was  provided  with  only  half  the  number,  which 
proved  to  be  amply  sufficient. 

It  was  necessary  to  readjust  the  height  of  the  board  G,  whenever  a  material 
change  was  made  in  the  depth  of  water  on  the  weir.  It  is  represented  in  figure 
5,  as  it  was  in  experiments  1  to  7,  in  which  the  depth  on  the  weir  was  near  the 
maximum.  The  top  of  the  board  G,  in  these  seven  experiments  was  about  0.105 
feet  below  the  top  of  the  weir. 

213.  The   depth   on   the   weir    was    observed    at    each    division    separately,  by 
means   of   hook    gauges,   similar   to    that   represented   by   figures   2,   3,   and   4,  plate 
XIII.     A  gauge  acting  on  the  same  principles  is  described  in  article  45.     The  gauge 
for   the  Westerly  division  was  placed  in  the  box  "ff,  figures  3,  4,  and  5,  plate  XVI. ; 
this   box   was   carefully   made    so    that   no   water   passed   into    or   out    of    it,   except 
through    the    pipes    in    the    bottom,   and    it   was    strongly   fastened   to   the    post  /, 
which   was   firmly   set   in   the    earth  at   the    bottom   and   supported  by  the  braces  K 
at    the    top.     When    observations   were    being   made   with   the   hook   gauge   for   the 
depth  on  the  weir,  the  three  pipes  L  L  L  formed  the  only  communication  between 
the   water  in   the   box   and    the    water   in   the   basin   between   the   grating   and    the 
weir;   the  surface  of  the  water  in  the  box  was  assumed  to  be  at  the  height  giving 
the  mean  depth  on  this  division  of  the  weir ;  subject,  however,  to  a  small  correction 
to  be  described  hereafter. 

214.  The  small  box  0  was  firmly  secured  to  the  planking  forming  the  interval 
between  the   two   divisions   of  the   weir;   it   had   no   communication   with    the   water 
outside  of  it,  except  by  means   of  the   pipes  JW  and   Q,  which  furnished  the  means 


IN   OPEN   CANALS   OF   UNIFORM   RECTANGULAR   SECTION. 


179 


of  connecting  it  with  either  of  the  hook  gauge  boxes  when  desired.  The  box  O 
contained  a  stationary  hook,  the  point  of  which  was  formed  by  a  portion  of  a 
sphere  of  about  half  an  inch  in  diameter;  the  coincidence  of  the  level  of  the  sur- 
face of  the  water  with  the  highest  part  of  the  spherical  surface  could  be  as  defi- 
nitely ascertained,  as  if  the  hook  had  terminated  in  a  sharp  point,  as  in  the  hook 
gauges,  whilst  the  spherical  surface  permitted  a  levelling-rod  to  be  placed  upon  it 
for  the  purpose  described  presently.  For  convenience  in  using  the  hook  gauges, 
their  zero  points  were  placed  several  inches  above  the  top  of  the  weir.  In  order  to 
ascertain  the  precise  elevation  of  the  zero  point  of  one  of  these  gauges  relatively  to 
the  mean  height  of  the  top  of  the  corresponding  division  of  the  weir,  the  water  was 
adjusted  to  a  depth  of  about  one  foot  on  the  weir,  the  three  pipes  L  were  closed, 
and  the  pipe  N  opened.  The  pipe  N  then  furnished  a  free  communication  between 
the  boxes  H  and  0,  neither  of  which  at  this  time  had  any  other  orifice  for  the 
passage  of  water  in  or  out.  Water  was  then  put  into  or  taken  out  of  these  boxes 
until  its  surface  coincided  with  the  highest  part  of  the  spherical  surface  which 
formed  the  point  of  the  stationary  hook  in  the  box  0;  when  this  was  done  and 
the  water  in  the  boxes  free  from  oscillations,  the  height  of  the  surface  of  the  water 
in  the  box  H  was  observed  by  means  of  the  hook  gauge,  which  evidently  gave  the 
height  of  the  point  of  the  stationary  hook  in  the  box  0,  by  the  scale  of  the -hook 
gauge  in  the  box  M.  The  height  of  the  point  of  the  stationary  hook  in  the  box 
O  above  the  mean  height  of  the  top  of  the  weir  was  obtained  by  levelling  with  a 
Troughton  and  Simms  dumpy  level ;  this  was  done  with  great  care  and  with  all 
the  precautions  necessary  for  insuring  accuracy ;  it  was  done  three  times  during  the 
course  of  the  experiments,  with  the  results  given  in  the  following  table. 

TABLE    XXI. 


Height  of  the  point  of  the 

Height  of  the  point  of  the 

Hook  in  the  Box  0  above 

Hook  in  the  Box  0  above 

Date. 

the  mean   height  of  the 

the   mean  height  of  the 

top  of  the  Westerly  divis- 

top of  the  Easterly  divis- 

ion of  the  Weir. 

ion  of  the  Weir 

1856. 

Feet. 

Feet. 

October        7. 

1.0087 

1.0112 

"           17. 

1.0090 

1.0111 

November  19. 

1.0089 

1.0127 

215.  From  the  observations  in  the  preceding  table  it  is  evident  that  the  rela- 
tive elevations  of  the  weir  and  the  point  of  the  stationary  hook  were  not  subject 
to  sensible  change.  Comparisons  between  the  hook  gauges  and  the  stationary  hook 
were  made  every  day,  with  a  depth  of  about  one  foot  on  the  weir,  and  the  cor- 


180  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

rection  determined  and  used  in  all  the  experiments  of  that  day.  The  relative 
heights  of  the  hook  gauges  and  stationary  hook  were  subject  to  greater  changes 
than  were  observed  between  the  stationary  hook  and  the  top  of  the  weir.  The 
experiments  extended  from  October  7  to  November  13;  the  difference  of  height 
of  the  stationary  hook  and  the  zero  of  the  Westerly  hook  gauge  was  greatest  on 
October  8,  when  it  was  0.4402  feet,  and  least  on  October  23,  when  it  was  0.4352 
feet,  the  change,  which  was  not  abrupt,  being  0.0050  feet.  The  corresponding  change 
at  the  Easterly  hook  gauge  was  0.0066  feet,  the  sign  and  dates  being  the  same  as 
at  the  Westerly  hook  gauge.  These  differences  are  not  very  great,  and  as  the  correc- 
tions were  determined  daily,  no  appreciable  errors  can  result  therefrom. 

216.  The  experiments  of  1852,  described  in  a  former  part  of  this  work,  from 
which  the  formula  for  computing  the  quantity  of  water  flowing  over  the  weir  in 
these  experiments  is  deduced,  were  made  upon  a  weir  of  great  simplicity  of  form,  in 
which  the  sheet  of  water  passing  over  the  weir  had  an  unobstructed  fall  of  not  less 
thon  three  feet;  see  figure  1,  plate  XIII.  Other  experiments  indicated  that  the 
sheet  of  water  may  meet  with  great  obstructions  soon  after  passing  the  weir,  with- 
out its  flow  over  the  weir  being  sensibly  affected  thereby  (see  ante,  page  134),  and 
it  was  thought,  that  in  these  experiments  the  obstructions  to  the  flow  of  the  water 
after  passing  the  weir,  would  affect  the  discharge  over  the  weir  to  so  small  an 
extent  as  to  be  inappreciable.  It  was  highly  important,  however,  to  avoid  all  ques- 
tion on  this  point;  and  to  determine  the  matter,  a  special  series  of  experiments  was 
undertaken. 

For  this  purpose  two  weirs  were  erected  in  the  upper  chamber  of  the  Lower 
Locks  in  Lowell,  K,  figure  1,  plate  XL  The  upper  weir  was  constructed  of  a  form 
to  which  the  formula  for  computing  the  discharge  could  be  applied  without  objec- 
tion. The  lower  weir  in  a  portion  of  the  experiments  was  of  the  same  form  as 
the  upper  weir,  and  in  the  other  portion  the  form  was  the  same  as  the  weir  at 
the  Tremont  Wasteway.  The  experiments  consisted  in  causing  the  same  volume  of 
water  to  flow  over  both  weirs,  and  observing  the  depth  assumed  by  the  water  on 
each  weir,  when  the  flow  had  become  permanent,  the  differences  in  the  depths,  if 
any,  being  due  to  differences  in  the  forms  and  conditions  of  the  two  weirs. 

The  Lock  chamber  is  twelve  feet  wide,  and  the  weirs  were  each  eight  feet  long, 
leaving  a  space  of  two  feet  at  each  end  to  insure  complete  contraction.  The  up- 
stream faces  of  the  weirs  were  vertical  planes,  and  the  crests  and  ends  were  of  the 
same  form  as  the  weir  at  the  Tremont  Wasteway.  The  bottoms  of  the  channels  on 
the  up-stream  sides  of  both  weirs  were  six  feet  below  the  tops  of  the  weirs.  The 
water  entered  the  Lock  chamber  through  the  head  gates,  and  under  a  head  of 
several  feet,  which  caused  a  great  commotion  in  the  water  at  the  upper  end  of 


IN  OPEN   CANALS   OF   UNIFORM   RECTANGULAR   SECTION.  181 

the  chamber.  The  upper  weir  was  placed  about  sixty  feet  from  the  upper  end  of 
the  chamber,  and  to  obliterate  the  disturbance  in  the  water  before  it  reached  the 
weir,  three  gratings,  at  right  angles  to  the  sides,  were  placed  across  the  chamber 
at  intervals  of  about  twelve  feet;  each  grating  contained  about  one  half  of  the 
aperture  per  square  foot,  for  the  passage  of  water,  as  the  grating  used  at  the 
Tremont  Wasteway.  The  lower  grating  was  about  fourteen  feet  from  the  weir.  The 
surface  of  the  water  between  the  two  upper  gratings  was  nearly  all  covered  by  a 
float  of  planks,  for  the  purpose  of  obliterating  the  oscillations  of  the  surface.  The 
second  or  lower  weir  was  about  thirty-five  feet  from  the  upper  weir,  and  similar 
arrangements  were  made  for  obliterating  disturbances  in  the  water,  as  were  provided 
for  the  upper  weir,  except  that  there  were  only  two  gratings,  the  disturbances 
caused  by  the  fall  of  the  water  from  the  upper  weir  into  the  basin  below  it  being 
much  less  than  were  caused  by  the  entrance  of  the  water  at  the  upper  end  of  the 
chamber.  The  lower  grating  was  about  fourteen  feet  from  the  lower  weir.  The 
depths  of  the  water  on  the  weirs  were  observed  by  means  of  hook  gauges  similar  to 
that  represented  on  plate  XIII.  The  difference  of  the  leakages  into  and  out  of  the 
part  of  the  chamber  included  between  the  two  weirs  was  ascertained,  and  a  cor- 
rection applied  for  the  same ;  and  also  for  the  rise  or  fall,  if  any,  of  the  surface  of 
the  water  in  the  same  space  during  the  time  occupied  by  an  experiment. 

In  arranging  the  apparatus,  it  was  designed  to  make  the  immediate  approach  of 
the  water  to  the  two  weirs  precisely  alike.  It  was  not  certain,  however,  that  the 
precautions  taken  to  insure  uniformity  would  produce  the  desired  result.  To  avoid 
doubts  on  this  point,  the  lower  weir  in  part  of  the  experiments,  as  stated  above,  was 
made  of  the  same  form  as  the  upper  weir,  in  which  case  any  difference  in  the 
depths  on  the  two  weirs,  the  quantity  of  water  flowing  being  the  same  at  both,  and 
there  being  no  obstructions  below,  must  be  due  to  differences  in  the  immediate 
approach  of  the  water  to  the  weirs.  A  series  of  experiments  was  made  under  these 
circumstances,  with  different  quantities  of  water  flowing,  from  which  it  was  ascer- 
tained, that  when  the  depth  on  the  upper  weir  was  about  0.5  feet,  the  depth  on 
the  lower  weir  was  0.0008  feet  greater;  when  the  depth  was  about  a  foot  on  the 
upper  weir,  it  was  the  same  on  the  lower  weir;  when  about  1.5  feet  on  the  upper 
weir,  it  was  0.0040  feet  less  on  the  lower  weir;  and  when  about  2  feet  in  depth  on 
the  upper  weir  it  was  about  0.0094  feet  less  on  the  lower  weir.  These  differences 
were  probably  due  to  small  differences  in  the  relative  velocities  of  the  water  imme- 
diately approaching  the  weirs,  at  different  depths,  and  might,  doubtless,  have  been 
partially  remedied  by  suitable  modifications  of  the  gratings.  It  would  have  required 
much  time,  however,  and  was  not  essential  to  our  arriving  at  correct  results,  the  ex- 
periments with  the  two  weirs  alike  having  been  sufficiently  numerous  and  varied  to 
enable  a  table  of  corrections  to  be  made. 


182  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

217.  Another   series   of  experiments  was   made   with   the   lower  weir  like   that 
erected  at  the   Tremont  Wasteway,  the  apron,  trough,  pipes,  standards,  etc.,  being  re- 
produced,  as   nearly  as   the   length   of  the  weir  would   permit.      The   height   of  the 
board,  forming  the  down-stream  side    of  the    trough,  was  of  course  varied   in  the  dif- 
ferent  experiments,   to   conform   to    the   corresponding  changes  at   the  Tremont -weir. 
The  upper  weir  remained  unchanged    throughout   all   the    experiments.     Water  being 
admitted   at   the   upper   end   of  the  chamber,  and   the  flow  become  permanent,  or  as 
nearly  so  as  practicable,  observations   were   made   of  the   depth   which   the   water  as- 
sumed  at   the    two   weirs.      It  would   occupy  much   space   to   describe  all  the   exper- 
iments made;    it  will  perhaps  be    sufficient   to    state    some    of  the    results   arrived  at. 
After  correcting  the  depth  on  the  lower  weir  for  the  differences  described  in  the  pre- 
ceding  section,  which   did  not  depend   on    the  forms   of  the  weirs,  the    following  dif- 
ferences  were    found.      When    the    depth   on  the    upper  weir  was  about  0.8  feet,  the 
depth  on  the  lower  weir  was  0.0007  feet  less ;    when  the  depth  on  the   upper  weir 
was  about  1.5  feet,  the  depth  on  the  lower  weir  was   the  same ;  •  when  the  depth  on 
the   upper   weir   was   about   2   feet,   the   depth   on   the   lower  weir  was   0.0085  feet 
greater.     This  last  difference  corresponds  to  a  diminution  of  flow  over  the  loAver  weir, 
with  the  same  depth  on  the  weir,  of  T^^. 

218.  The   effect  of  what   appear  to  be  obstructions  to  the  flow  over  a  weir  is, 
generally,  to   increase  the  depth  on  the  weir  over  what  it  would  be  if  the  flow  was 
free ;    sometimes,  however,  it  has  the  contrary  effect.     (See  article  137.)     The  exper- 
iments at  the  Lower  Locks  described  in  the  preceding  section  furnished  the    data  for 
a   table   of  corrections   of  the   depths    of  water   on   the  Tremont  weir,  due   to   the 
obstructions  to  the  flow  of  the  sheet  after  passing  the   crest  of  the  weir.      In  exper- 
iment  1,   table  XXII.,  in  which   this   correction   has  nearly   its   greatest  value,  it  is 
—  0.0058  feet 

219.  Another   small   correction   was   also   applied.     In  the  experiments  of  1852 
(art.  173),  it  was   found  that  there  was  no  sensible  difference  in  the  observed  depth 
upon   the   weir,  whether   the  external  orifice  of  the  pipe,  forming  the  communication 
between  the  water  approaching  the   weir  and    the   hook    gauge   box,  was   close   to 
the   plane  of  the  weir   or  six  feet  up  stream   from   that  plane,  the   external   orifice 
of  the   pipe   being   at  a   considerable   depth   below  the  top   of  the  weir.     In  arrang- 
ing   the    apparatus    at    the   weir    at    the   Tremont   Wasteway,   it  was   thought  that 
there   would   be   less  liability   to   errors  in  the  observed  depths,  from  currents  acting 
on   the   external  orifices    of   the   pipes,   if   they  were   very  near   the   plane   of   the 
weir,   and    at    the    bottom    of   the   canal,   and    they   were    accordingly   so   arranged. 
In    the    experiments    of   1852,  however,   on   which    the   formula  for  computing  the 
flow  over   the   weir  is  founded,  the   orifice   in   the    hook    gauge    box  was   six  feet 


IN  OPEN  CANALS  OF  UNIFORM  RECTANGULAR  SECTION.  183 

from  the  weir,  and  in  order  to  ascertain  whether  any  difference  could  be  detected 
in  the  observed  depths  on  the  weir  at  the  Tremont  Wasteway,  with  the  external 
orifice  of  the  pipe  at  different  distances  from  the  weir,  some  special  experiments 
were  made. 

For  this  purpose  an  apparatus  of  pipes  similar  to  that  represented  in  figures 
8  and  9,  plate  XIV.,  was  placed  at  the  bottom  of  the  canal,  on  the  up-stream 
side  of  the  weir  at  the  Tremont  Wasteway.  The  orifices  of  the  pipes  were  pro- 
tected from  the  action  of  lateral  currents,  if  any  existed,  by  a  second  board, 
placed  parallel  to  the  board  in  which  the  lower  ends  of  the  pipes  were  inserted, 
and  three  inches  distant;  these  boards  were  placed  at  right  angles  to  the  weir, 
and  the  space  between  them  was  open  at  the  top  and  the  up-stream  end,  so 
that  the  current  flowing  towards  the  weir,  flowed  through  the  trough  formed  by 
the  two  boards,  by  the  open  ends  of  the  pipes,  which,  to  avoid  eddies,  did  not 
project  beyond  the  plane  of  the  board.  With  this  apparatus,  observations  were 
made  of  the  differences  in  the  depths  on  the  weir,  when  the  different  pipes  were 
in  communication  with  the  hook  gauge  box ;  substantially  the  same  precautions 
being  taken  to  secure  precision  in  the  results  as  are  described  in  article  170. 

Taking  the  observations  made  with  the  pipe  opening  at  0.52  feet  from  the 
weir,  as  represented  at  B,  figures  3,  4,  and  5,  plate  XVI.,  as  the  standard ;  when 
the  depth  on  the  weir  was  about  0.76  feet,  the  differences  in  the  depths  observed 
by  means  of  the  other  pipes  were  as  follows :  — 

By  the  pipe  opening  at  2  feet  from  the  plane  of  the  weir,  difference  =  —  0.0003  feet. 

«  «  «          4         «  «  «  «  «          — 0.0003    " 

a          «  «         e        «  «  «          «  «         —  —  0.0004    " 

a  it  «  g  "  "  "  «  "  =  —  0.0001     « 

"          «             "       10        "               "              "          "           "         =  —  0.0003    " 
a          «  "       12        "  "  "          "  "         — 0.0012    " 

When  the  depth  on  the  weir  was  about  1.44  feet,  the  differences  observed 
were  as  follows :  — 

By  the  pipe  opening  at  2  feet  from  the  plane  of  the  weir,  difference  =  -f-  0.0020  feet. 

a  u  u  4  «  «  «  «  «  —  Q  0009     " 

«  «  «  6  "  "  "  "  "  ==  —  0.0013  u 

u  u  u  g  «  «  «  «  «  _  —  0.0054  " 

«  «  «  10  «  «  «  «  «  =  —  0.0089  " 

«  «  «  12  "  "  "  "  "  = 0.0124  " 

Up  to  six  feet  from  the  weir,  these  differences  are  very  small;  it  was  thought 
best,  however,  to  take  account  of  them. 


184  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

By  a  discussion  of  the  whole  of  the  experiments  a  table  was  formed,  for  cor- 
recting the  observed  depths  on  the  weir,  to  what  they  would  have  been  if  observed 
with  the  pipe  opening  at  G  feet  from  the  weir. 

When  the  depth  on  the  weir  is  0.5  feet,  this  correction  is  —  0.0002  feet. 

a                 «  «          «   0.8           "                         "    —  0.0004    " 

«  "   1.0           «                         «    —  0.0006    " 

»                *  «   1.5          "                        "    —  0.0014    " 

«                  «  "   2.0           "                         "    —  0.0023    " 

220.  By   table    XVIII.,   containing    the    results   of    similar    experiments    at    the 
Lower   Locks,   made   about   four  years    previously,   it    will    be   seen,   that   the    differ- 
ences  between   the   depths   on   the   weir,   observed   by   means   of  a   pipe  opening   at 
six   feet    from    the    plane   of    the   weir,   and   by   a    pipe  opening    at    one   inch    from 
the    plane  of  the  weir   (changing   the    signs   to   conform   to  the    experiments   at   the 
Tremont  weir),   were    as   follows:  — 

When  the  depth  on  the  weir  was  about  0.80  feet,  difference  =  —  0.00060  feet. 

"  1.00      "  "  -  0.00033    " 

The  small  differences  in  these  results  from  those  obtained  at  the  Tremont 
Wasteway  weir  may  be  explained  by  the  different  forms  of  the  approaches  to  the 
weirs,  and  the  different  arrangement  of  the  apparatus. 

221.  The    formula   for   computing   the  quantity  of  water  flowing  over  weirs,  de- 
duced from  the  experiments  made  at  the  Lower  Locks  in  1852,  viz. : 

Q  =  3.33  (L  —  0.1  n  H}  H%,  (A.) 

is  adapted  to  weirs  of  widely  differing  proportions,  including  all  the  forms  on  which 
experiments  are  given  in  table  XIII.  By  reference  to  column  16  in  that  table,  it 
will  be  seen,  however,  that  the  experiments  on  each  particular  description  of  weir 
generally  give  a  coefficient  differing  slightly  from  the  mean  value  deduced  from 
the  whole  of  the  experiments.  In  case  any  of  those  particular  forms  should  be 
reproduced,  it  is  evident,  that  the  quantity  of  water  flowing  over  the  same  could 
be  more  accurately  computed,  by  using  the  corresponding  coefficient  given  in 
column  16,  than  by  using  that  given  in  formula  (A.),  which  is  a  mean,  deduced 
from  the  whole  of  the  experiments.  In  determining  the  formula  by  which  to  com- 
pute the  flow  over  the  weir  at  the  Tremont  Wasteway,  it  was  apparent  that 
results  more  exact  could  be  attained  by  deducing  a  new  formula  from  a  selection 
of  the  experiments  given  in  table  XIIL,  in  which  the  circumstances  were  most 


IN   OPEN   CANALS   OF  UNIFORM   RECTANGULAR   SECTION.  185 

nearly  like   those   at  the  Tremont  Wasteway  weir.     For  this  purpose  53  experiments 
were  selected,  and  the  formula  deduced  from  them  is 

Q  =  3.318  (L  —  0.08  n  H)  H%.  (B.} 

As  applied  to  the  weir  at  the  Tremont  Wasteway, 

When  the  depth  is  1  foot,  the  discharge  by  formula  (A.)  =  265.09  cubic  feet  per  sea 

And  by  formula  (B.) —  264.40       «        "      «       « 

Difference   ^ =      0.69       «        «      «       « 

When  the  depth  is  2  feet,  the  "discharge  by  formula  (A.)  =  746.02       «        "      " 

And  by  formula  (B.) =  744.84       «        «      « 

Difference    ?fa =      1.18       «        "      " 

When  the  depth  is  3.5527  feet,  both  formulas  give  the  same  discharge. 

222.  In  making  these  experiments,  there  were  several  objects  in  view,  which 
may  be  classed  under  two  heads,  viz. :  — 

1st.  To  determine  a  formula  for  correcting  the  quantity  passing  a  measuring 
flume,  as  deduced  from  the  mean  velocity  of  the  tubes;  there  being  no  unusual 
disturbing  causes. 

2d.  To  ascertain  the  degree  of  uniformity  in  measurements  made  under  like 
circumstances ;  and  to  determine  the  magnitude  of  the  errors  to  which  we  are 
liable,  when  measurements  are  made  under  exceptionable  circumstances,  such  as  high 
winds  and  great  irregularities  in  the  motion  of  the  water. 

The  experiments  adapted  to  the  first  object  were  necessarily  made  under  the 
normal  conditions  of  freedom  from  high  wind,  and  from  great  irregularity  in  the 
currents.  Table  XXII.  contains  105  experiments  selected  as  being  suitable  for 
this  purpose,  and  table  XXV.  contains  35  experiments  made  for  the  purposes  in- 
cluded in  the  second  class. 


24 


186 


TABLE 
EXPERIMENTS  MADE  AT  THE  TREMONT  WEIR  AND  MEASURING  FLUME, 


1 

3 

3 

Weir  Measurement. 

Flume 

Tempera- 

4 

5 

6 

7 

8 

9 

1O 

11 

IS 

13 

14 

15 

tureVin 
degrees  of 
Fahrenheit's 

Quantity 
of  water 

Corrected 
quantity 

Difference 
between 
the  depth 

Date. 

thermometer 

Total 

Observed 

Observec 

Mean 

Corrected 

passing 
over  the 

Correc- 

passing 
the  flume. 

Mean 

Length 

of  water 
in  the 

No. 

length 
of  the 
weirs. 

depth  of 
water  on 
the 

depth  of 
water  on 
the 

observed 
depth  of 
water 

depth  of 
water 
on  the 

weirs, 
com- 
puted 

tion  for 
the  leak- 
age into 

deduced 
from  the 
weir 

Mean 

width  of 
the  flume. 

depth  of 
water 
in  the 

of  the 
im- 

mersed 

flume  and 
the  length 
of  the 

the 

1866. 

of  the 

Westerly 
weir. 

Easterly 
weir. 

on  the 
weirs. 

weirs. 

by  the 
formula 

the  Hume 

measure- 
ment. 

fluuie. 

part 
of  the 

immersed 
part  of 

atmos- 

of the 

tube. 

the  tube, 

Exp 

phere 

water. 

£ 

H 

Q  — 

Q' 

divided  by 

shade. 

3.318 

(t—  O.OSn//)//"' 

the  depth 
of  water  in 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Cubic  ft. 

Cubic  ft. 

Cubic  fl. 

Feet. 

F«et. 

Feet. 

the  flume. 

per  sec. 

per  sec. 

per  sec. 

D 

1 

Oct.     7    A.M. 

52.5 

57.0 

80.007 

1.8972 

1.8585 

.8778 

1.8839 

681.25 

0 

681.25 

26.746 

9.533 

9.482 

0.005 

2 

it        11       u 

53.5 

57.0 

tt 

1.8750 

1.8395 

.8572 

1.8634 

670.U2 

0 

670.22 

tt 

9.515 

9.430 

0.009 

3 

11       a       u 

60.5 

57.0 

tt 

1.8556 

1.8214 

.8385 

1.8448 

660.26 

0 

660.26 

tt 

9.496 

9.380 

0.012 

4 

tt       tt       tt 

64.5 

57.5 

tt 

1.8720 

1.8362 

.8541 

1.8604 

668.61 

0 

668.61 

tt 

9.510 

9.330 

0.019 

i     5 

"       "    P.M. 

65.0 

58.0 

ti 

1.8906 

1.8548 

.8727 

1.8787 

678.45 

0 

678.45 

tt 

9.531 

9.230 

0.032 

6 

tt       tt       u 

65.0 

58.0 

ti 

1.8896 

1.8539 

.8717 

1.8777 

677.91 

0 

677.91 

tt 

9.532 

9.130 

0.042 

7 

ti       it       ti 

64.0 

58.0 

it 

1.8872 

1.8507 

.8689 

1.8750 

676.45 

0 

676.45 

tt 

9.530 

8.530 

0.105 

8 

"        8    A.M. 

51.5 

57.0 

80.008 

1.7846 

1.7527 

1.7686 

1.7752 

623.43 

0 

623.43 

tt 

9.422 

9.360 

0.007 

9 

u        it       ti 

60.5 

57.0 

it 

1.7882 

1.7570 

1.7726 

.7790 

625.42        0 

625.42 

tt 

9.426 

9.320 

0.011 

10 

tl        tt       ft 

63.5 

57.0 

it 

1.7825 

1.7495 

1.7660 

.7726 

622.07;        0 

622.07 

it 

9.421 

9.280 

0.015 

11 

"     ""     P.M. 

66.6 

57.0 

tt 

1.7810 

1.7490 

.7650 

.7716 

621.54        0 

621.54 

it 

9.421 

9.220 

0.021 

12 

fl        tf       tt 

66.1 

57.0 

tt 

1.7720 

1.7416 

.7568 

.7633 

617.20        0 

617.20 

tt 

9.412 

9.120 

0.031 

13 

tf        tt       tt 

62.9 

57.2 

tt 

1.7713 

1.7394 

.7553 

.7618 

616.4l|        0 

616.41 

it 

9.410 

9.020 

0.041 

14 

ft        tt       tt 

60.0 

58.0 

tt 

1.7626 

1.7333 

.7479 

.7546 

612.66 

0 

612.66 

tt 

9.402 

8.410 

0.106 

15 

"        9    A.M. 

53.0 

58.0 

80.009 

1.5061 

1.4864 

.4962 

.5027 

486.08 

0 

486.08 

tt 

9.141 

9.080 

0.007 

16 

tt        it       it 

59.5 

58.0 

tf 

1.5045 

1.4854 

.4949 

.5015 

485.50        0 

485.50 

tt 

9.140 

9.030 

0.012 

17 

ti        it       tt 

70.5 

58.0 

H 

1.5046 

1.4849 

.4947 

.5013 

485.40 

0 

485.40 

ti 

9.138 

8.980 

0.017 

18 

"        "     P.M. 

80.5 

58.0 

tt 

1.5038 

1.4842 

.4940 

.5006 

485.06 

0 

485.06 

ti 

9.138 

8.930 

0.023 

19 

tt        tt       tt 

79.5 

58.5 

ft 

1.5033 

1.4833 

.4933 

.4999 

484.72 

0 

484.72 

tt 

9.137 

8.830 

0.034 

20 

tt        tt       it 

79.5 

59.0 

tt 

1.4925 

1.4737 

.4831 

.4895 

479.71 

0 

479.71 

u 

9.126 

8.730 

0.043 

21 

tt        it       tt 

74.0 

59.0 

tt 

1.4839 

1.4639 

.4739 

.4804 

475.34 

0 

475.34 

tt 

.9.118 

8.120 

0.109 

22 

"      11    A.M. 

72.5 

59.0 

80.010 

1.2091 

1.1995 

.2043 

.2086 

351.03 

0 

351.03 

26.745 

8.838 

7.830 

0.114 

23 

tt        it        it 

78.5 

59.0 

tt 

1.2131 

1.2031 

.2081 

.2124 

352.68        0 

352.68 

fl 

8.842 

8.430 

0.047 

24 

"        "     P.M. 

78.0 

60.0 

tt 

1.1968 

1.1899 

.1933 

.1975 

346.22        0 

346.22 

tf 

8.830 

8.530 

0.034 

25 

ft               ft               tt 

77.5 

60.0 

» 

1.1942 

1.1857 

.1899 

.1941 

344.75        0 

344.75 

ft 

8.827 

8.630 

0.022 

26 

"      13   A.M. 

57.5 

60.0 

80.011 

1.1985 

1.1886 

.1935 

.1977 

346.31         0 

346.31 

tf 

8.827 

8.780 

0.005 

27 

tt        tt       tt 

59.0 

60.0 

tt 

1.1854 

1.1759 

.1806 

.1847 

340.70        0 

340.70 

ft 

8.815 

8.730 

0.010 

28 

tt        tt        tt 

64.5 

60.0 

tt 

1.1820 

1.1726 

.1773 

.1813 

339.24 

0 

339.24 

ft 

8.810 

8.680 

0.015 

29 

"        vl     P.M. 

69.0 

60.0 

tt 

1.3715 

1.3555 

.3635 

.3691 

422.96 

0 

422.96 

If 

8.997 

7.980 

0.113 

30 

tt        tt       tt 

68.0 

60.0 

tt 

1.3533 

1.3382 

.3457 

.3512 

414.72         0 

414.72 

ft 

8.981 

8.600 

0.042 

31 

it        tt       ft 

67.0 

60.0 

it 

1.3580 

1.3427 

.3503 

.3559 

416.87        0 

416.87 

ft 

8.985 

8.700 

0.032 

32 

tt        tt       tt 

65.5 

60.0 

tt 

1.3524 

1.3387 

.3455 

.3510 

414.63        0 

414.63 

ft 

8.978 

8.800 

0.020 

33 

"      14   A.M. 

40.0 

59.0 

80.012 

1.3752 

1.3618 

.3685 

.3742 

425.32        0 

425.32 

ft 

9.006 

8.850 

0.017 

34 

tt        it       tt 

40.5 

59.0 

it 

1.3772 

1.3635     .3703 

.3760 

426.15 

0 

426.15 

(f 

9.009 

8.900 

0.012 

35 

tt        tt       tt 

42.0 

59.0 

tt 

1.3681 

1.3552     .3616 

.3673 

422.13 

0 

422.13 

tt 

8.997 

8.960 

0.004 

36 

"        "     P.M. 

45.0 

58.0 

tt 

0.9792 

0.9756   0.9774 

0.9801 

256.58 

0 

256.58 

ft 

8.609 

8.230 

0.044 

37 

tt        ti       tt 

43.5 

58.0 

tt 

0.9840 

0.9818   0.9829 

0.9856 

258.74 

0 

258.74 

(t 

8.615 

8.330 

0.033 

38 

tt        tt       tt 

41.5 

58.0 

tt 

0.9746 

0.9732 

0.9739 

0.9766 

255.21 

0 

255.21 

tt 

8.604 

8.430 

0.020 

39 

"      15   A.M. 

34.5 

56.0 

u 

1.0016 

0.9971 

0.9993 

1.0022 

265.29 

0 

265.29 

tt 

8.631 

8.480 

0.017 

40 

tt               tt              tt 

39.0 

56.0 

tt 

0.9916 

0.9872 

0.9894 

0.9922 

261.34 

0 

261.34 

if 

8.620 

8.530 

0.010 

41 

tf               tf              ft 

47.5 

56.0 

tt 

0.9841 

0.9805 

0.9823 

0.9850 

258.51 

0 

258.51 

ft 

8.611 

8.570 

0.005 

42 

"        "     P.M. 

52.5 

56.0 

tt 

0.9942 

0.9900 

0.9921 

0.9950 

262.44 

0 

262.44 

(t 

8.626 

7.620 

0.117 

43 

tt        tt        tt 

50.0 

56.0 

tt 

0.5504 

0.5502 

0.5503 

0.5513 

108.43 

0 

108.43 

(f 

8.172 

7.120 

0.129 

44 

"      16    A.M. 

38.5 

54.0 

80.014 

0.5445 

0.5444 

0.5444 

0.5454 

106.70 

0 

106.70 

tf 

8.167 

7.720 

0.055 

45 

it        it       u 

47.0 

54.0 

tl 

0.5403 

0.5406 

0.5404 

0.5414 

105.53 

0 

105.53 

(f 

8.164 

7.920 

0.030 

46 

if        ti        tt 

55.5 

54.0 

ft 

0.5394 

0.5395 

0.5394 

0.5404 

105.24 

0 

105.24 

(f 

8.163 

8.070 

0.011 

47 

"        "     P.M. 

60.5 

54.0 

fl 

0.5410 

0.5412 

0.5411 

0.5421 

105.74 

0 

105.74 

tf 

8.165 

8.122 

0.005 

48 

ft        ft       it 

58.5 

54.0 

tt 

0.5342 

0.5350 

0.5346 

0.5356 

103.84 

0 

103.84 

ft 

8.159 

8.020 

0.017 

49 

.i     21     " 

71.0 

53.5 

tt 

0.5447 

0.5454 

0.5450 

0".5460 

106.88 

0 

106.88 

ft 

8.171 

8.070 

0.012 

50 

"      27  A.M. 

35.0 

47.0 

ft 

1.9105 

1.8662 

1.8883 

1.8943 

686.93 

0 

686.93 

<t 

9.540 

9.380 

0.017 

51 

ti      tt      tt 

36.0 

47.0 

ft 

1.9152 

1.8714 

1.8933'  1.8992   689.58        0 

689.58 

fl 

9.548 

9.330 

0.023 

52 

it      ti      tt 

39.0 

47.0 

ft 

1.9106 

1.8659 

1.8882!  1.8941'  686.82        0 

686.82 

tt 

9.543 

9.130 

0.043 

53 

u      ti      it 

39.0 

47.0 

tf 

1.8879 

1.84531  1.8666 

1.8726   675.22        0 

675.22 

tf 

9.521 

8.532 

0.104 

54 

it      11      it 

ft 

1.8909 

1.8484;  1.8696 

1.8757   67C.89 

0 

676.89 

tf 

9.525 

9.476 

0.005 

55 

ti      u      u 

57.0 

47.0 

ft 

1.8742 

1.8320   1.8531 

1.8593   668.07 

0 

668.07 

ft 

9.508 

9.230 

0.029 

56 

"        "     P.M. 

59.5 

48.0 

If 

1.8758 

1.8334    1.8546 

1.86091  668.94 

0 

668.94 

tf 

9.510 

9.431 

0.008 

XXII. 

FROM  WHICH  THE  FORMULA  OF  CORRECTION  C-- 


187 


0.116  (<JD  —  0.1)  IS  DETERMINED. 


Measurement. 

18 

19 

30 

31 

33 

23 

i  ft 

1w 

Difference 
between  the 

Proportion- 
al differ- 

Quanti- 
ty of 

Proportion- 
al differ- 

Remarks on  the  Force  and  Direction 

lo 

t 

quantity  of 

ence,  or 

water 

ence  of  the 

of  the  Wind  at  the  Flume,  during 

Quanti- 

water paas- 

the  differ- 

passing 

corrected 

the  Experiments. 

Mean 

ty  of 

ing  the  tlume 
de  luced  from 

ence  in  the 

the 

quantity  as 

No. 

velocity 
of  the 

passing 

the  mean 

column, 

deduced 

in  the 

of 

tubes 

the 

velocity  of 

divided  by 

from  the 

flume  given 

thvotrjh* 

flume, 

the  tubes, 

the  quan- 

mean 

in  the 

General  Remarks. 

the 

deduced 

and  the 

tity  de- 

velocity 

preceding 

nume, 

from  the 

quantity 

duced  from 

of  the 

column, 

Exp. 

by  the 

mean 
velocity 

deduced  from 
the  weir 

the  Hume 
measure- 

tubes 
correc'd 

and  the 
weir  meas- 

Force. 

Direction. 

A0   .     1. 

of  the 

measure- 

ment. 

by  the 

urement. 

tubes. 
f-tti 

ment. 

formula 

Q'"  —  Q' 

V 

Q"  —  Q' 

cyi 

Q'     = 

ty 

Feet 

Cubic  ft. 

Cubic  ft. 

per  sec. 

per  sec. 

JU-I-  SVC. 

Q'~(l—  u.liufv'i' 

-0.1)). 

1 

2.683 

iisi.ii; 

+  2.91 

+0.0043 

iiSH.49 

4-0.0077 

Moderate. 

Down  stream. 

Reduced  copies   of  the  diagrams,  constructed 

2 

2.616 

665.86 

—  4.36 

—0.0065 

666.26 

—0.0059 

n 

ti            it 

for  the  purpose  of  obtaining  the  mean  velocities 

3 

2.636 

669.55 

+  9.29 

+0.0139 

668.81 

+0.0129 

irregular. 

of  the  tubes,  in  experiments  1  and  7,  are  given 
n  plate  XVII. 

4 

2.667 

678.27 

+  9.66 

+0.0142 

675.29 

+0.0100 

ii 

tt 

5 

2.649 

675.23 

—  3.22 

—0.0048 

669.05 

—0.0139 

Very  gentle. 

ti 

6 

2.713 

691.72 

+13.81 

+0.0200 

683.30 

+0.0080 

Gentle. 

Down  stream. 

7 

2.726 

694.86 

+18.41 

+0.0265 

676.80 

+0.0005 

Hardly  perceptible. 

it         tt 

8 

2.451 

617.69 

—  5.74 

—0.0093 

618.86 

—0.0073 

«             tt 

if            it 

s 

9 

2.461 

620.49 

—  4.93 

—0.0079 

620.14 

—0.0084 

u            it 

tt            ti 

„ 

10 

2.473 

623.01 

-  0.94 

+0.0015 

621.38 

—0.0011 

ti                ti 

it              it 

11 

2.485 

626.18 

-  4.64 

-t-0.0074 

622.92 

-  -0.0022 

Gentle. 

tt             it 

12 

2.487 

626.14 

-  8.94 

+0.0143 

620.62 

-  -0.0055 

Very  gentle. 

ti             tt 

13 

2.484 

625.10 

-  8.69 

+0.0139 

617.67 

-  -0.0020 

Hardly  perceptible. 

it            it 

14 

2.525 

634.95 

-22.29 

+0.0351 

618.33 

-  -0.0093 

Calm. 

15 

1.984 

485.12 

—  0.96 

—0.0020 

486.04 

—0.0001 

Hardly  perceptible. 

Down  stream. 

16 

1.943 

474.94 

—10.56 

—0.0222 

474.41 

—0.0228 

Calm. 

17 

1.967 

480.68 

—  4.72 

—0.0098 

478.99 

—0.0132 

{{ 

18 

1.972 

481.97 

—  3.09 

—0.0064 

479.08 

—0.0123 

Hardly  perceptible. 

Down  stream. 

19 

2.003 

489.56 

-  4.84 

-  -0.0099 

484.77 

+0.0001 

Calm. 

20 

1.985 

484.54 

-  4.83 

-  -0.0100 

478.51 

—0.0025 

tt 

21 

2.025 

493.77 

-18.43 

-  -0.0373 

480.59 

+0.0110 

.« 

22 

1.536 

363.17 

-12.14 

-  -0.0334 

353.16 

+0.0061 

Very  gentle. 

Down  stream. 

23 

1.518 

359.06 

-  6.38 

-  -0.01  78 

354.20 

+0.0043 

(i        u 

tt            it 

24 

1.483 

350.13 

-  3.91 

-  -0.0112 

346.70 

+0.0014 

it          ti 

ti            tt 

25 

1.462 

345.25 

r  0.50 

-  -0.0014 

343.31 

—0.0042 

Moderate. 

it            ii 

26 

1.453 

343.03 

—  3.28 

—0.0096 

344.20 

—0.0061 

Calm. 

27 

1.436 

338.56 

—  2.14 

—0.0063 

338.56 

—0.0063 

Very  gentle. 

Down  stream. 

28 

1.446 

340.68 

+  1.44 

+0.0042 

339.79 

+0.0016 

Moderate. 

ii         ti 

29 

1.786 

429.81 

+  6.85 

-  -0.0159 

418.04 

—0.0116 

(  Moderate,  some-  ) 
I  times  calm.         / 

ii         tt 

30 

1.730 

415.58 

+  0.86 

--0.0021 

410.52 

—0.0101 

Hardly  perceptible. 

tt            t* 

31 

1.737 

417.33 

+  0.46 

--0.0011 

413.51 

—0.0081 

Moderate. 

ii            it 

32 

1.723 

413.69 

—  0.94 

—0.0023 

411.70 

—0.0071 

Hurdly  perceptible. 

tt            ti 

I!:; 

1.779 

428.60 

-  3.28 

+0.0077 

427.09 

+0.0042 

Brisk  but  variable. 

ti            ti 

34 

1.772 

126.99 

-  0.84 

+0.0020 

426.52 

+0.0009 

{Strong  but  va- 
riable. 

Generally  across. 

35 

1.762 

424.09 

-  1.96 

+0.0046 

425.90 

+0.0089 

tt         it       it 

tt            ti 

36 

1.138 

261.92 

-  5.34 

+0.0204 

258.59 

+0.0078 

ti         tt       tt 

ti            it 

37 

1.130 

260.38 

-  1.64 

-j-0.0063 

257.91 

—0.0032 

tt          tt       u 

tt                      It 

38 

1.110 

255.47 

-  0.26 

+0.0010 

254.24 

—0.0038 

U                tt           It 

tt            tt 

39 

1.155 

266.68 

-  1.39 

+0.0052 

265.74 

+0.0017 

Very  moderate. 

Irregular. 

40 

1.118 

257.81 

—  3.53 

—0.0137 

257.81 

—0.0135 

tt            » 

y 

41 

1.135 

261.50 

+  2.99 

+0.0114 

262.39 

+0.0150  Moderate. 

(  Irreg.,butgen, 
|  up  stream. 

42 

1.167 

269.18 

+  6.74 

+0.0250 

261.62 

—  0.0031  Very  moderate. 

Irregular. 

43 

0.519 

113.39 

+  4.96 

+0.0437 

109.98 

-  -0.0143  Hardly  perceptible. 

41 

0.498 

108.76 

+  2.06 

-  -0.0189 

107.06 

-  -0.0034  Calm. 

45 

0.497 

108.52 

+  2.99 

-  -0.0276 

107.60 

-  -0.01  96jVery  moderate. 

Generally  across. 

46 

47 

0.500    109.18 
0.495  1108.09 

: 

-  3.94 
-  2.35 

-  -0.0361  109.12 
-  -0.021  7  108.4U 

-  -0.0369 

-  -0.025  7 

tt        u 
(  Moderate,  some- 
(  times  calm. 

Irregular. 
Down  stream. 

48 

0.486    106.01 

-  2.17 

-  -0.0205  105.64 

-  -0.01  73 

Hardly  perceptible. 

tt         ti 

49 

0.497 

108.52 

-  1.64 

-  -0.0151 

108.40 

-  -0.0142 

Calm. 

50 

2.701 

689.07 

-  2.14 

-  -0.0031 

686.64 

—0.0004 

u 

51 

2.707 

691.16 

-  1.58 

-  -0.0023  687.02 

—0.0037 

Hardly  perceptible. 

52 

2.774 

708.05 

+21.23 

-  -0.0300 

699.23 

-  -0.0181 

Calm. 

53 

2.737 

696.89 

+21.67 

-  -0.03  11 

678.90 

-  -0.0055 

« 

54 

2.654 

676.22 

—  0.67 

—0.0010678.52 

-  -0.0024 

tt 

55 
56 

2.658 
2.596 

676.02 
660.36 

+  7.95 
—  8.58 

+0.0118J670.51 
—  0.0130i661.17 

-  -0.0037 
—0.0116 

" 

188 


TABLE 
EXPERIMENTS  MADE  AT  THE  TREMONT  WEIR  AND  MEASURING  FLUME, 


Weir  Measurement. 

Flume 

1 

a 

3 

Tempera- 

4 

5 

6 

V 

8 

9 

10 

11 

12 

13 

14 

15 

No. 

Date. 

ture,  in 
degrees  of 
Fahrenheit's 
thermometer, 

Total 
length 
of  the 

Observed 
depth  of 
water  on 
the 

Observed 
depth  of 
water  on 
the 

Mean 
observed 
depth  of 
water 

Corrected 
depth  of 
water 
on  the 

Quantity 
of  water 
passing 
over  the 
weirs, 
com- 
puted 

Correc- 
tion for 
the  leak- 
age into 

Corrected 
quantity 
passing 
theflume, 
deduced 
from  the 
weir 

Mean 
width  of 

the  1  1  ii  11  H  •  . 

Mean 
depth  of 
water 
in  the 

Length 
of  the 
im- 
mersed 

Difference 
between 
the  depth 
of  water 
in  the 
luii'c  and 
the  length 
of  the 

of 

1856. 

of  the 

Westerly 
weir. 

Easterly 
weir. 

on  the 
weirs. 

weirs. 

by  the 
formula 

the  Hume. 

measure- 
ment. 

flume. 

part 
of  the 

immersed 
part  of 

the 

atmos- 

of  the 

tube. 

the  tube, 

•vn 

phere 

water. 

L 

H 

Q  = 

Q' 

divided  by 

iutp. 

in 
shade. 

3.31! 

(£-0.08nfl 

the  depth 
of  water  in 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Cubic  ft. 

Cubic  ft. 

Cubic  ft. 

Feet. 

Feet. 

Feet. 

;he  tlume. 

per  sec. 

per  sec. 

per  sec. 

D 

57 

Oct.  27    P.M. 

58.0 

48.0 

80.014 

1.5186 

1.4955 

1.5070 

1.5137 

491.42 

0.00 

491.42 

26.745 

9.151 

!).(.!!)  7 

0.006 

58 

it      it      tt 

tt 

1.5011 

1.4797 

1.4904 

1.4969 

483.31 

0.00 

483.31 

ti 

9.135 

9.047 

0.010 

59 

tt       tt      tt 

54.0 

48.0 

tt 

1.5014 

1.4794 

1.4904 

1.4969 

483.31 

0.00 

483.31 

tl 

9.133 

8.850 

0.031 

60 

"      28   A.M. 

49.0 

48.0 

tt 

1.5204 

1.4981 

1.5092 

1.5157 

492.40 

0.00 

492.40 

2G.746 

9.154 

8.747 

0.044 

61 

ft       it       ii 

tt 

1.5212 

1.5007 

1.5109 

1.5175 

493.28 

0.00 

493.28 

" 

9.158 

9.000 

0.017 

62 

tt         ti        ft 

51.0 

48.0 

tt 

1.5210 

1.4995 

1.5102 

1.5168 

492.94 

0.00 

492.94 

tt 

9.157 

9.050 

0.012 

63 

tt         ii        tt 

51.0 

48.0 

tt 

1.5073 

1.4867   1.4970 

1.5035 

486.50 

0.00 

486.50 

tt 

9.145 

8.150 

0.109 

64 

Nov.  10  A.M. 

29.0 

44.0 

tt 

0.6888 

0.6872   0.6880 

0.6894 

151.55 

—0.14 

151.41 

13.372 

8.305 

8.150 

0.019 

65 

tt        it       it 

tt 

0.6861 

0.6858   0.6859 

0.6873 

150.86 

—0.14 

150.72 

fi 

8.301 

8.000 

0.036 

66 

tt        it        tt 

ti 

0.6850 

0.6844   0.6847 

0.6861 

150.46 

—0.14 

150.32 

ft 

8.298 

7.900 

0.048 

67 

tt        it       it 

37.0 

44.0 

tt 

0.6831 

0.6821    0.6826 

0.6840 

149.77 

—0.14 

149.63 

fi 

8.299 

8.250 

0.006 

68 

it        ft       tt 

tt 

0.6806 

0.6801    0.6803 

0.6817 

149.02 

—  0.14 

148.88 

if 

8.295 

8.100 

0.024 

69 

it        ft       tt 

42.0 

44.0 

tt 

O.G781 

0.6777   0.6779 

0.6793 

148.24 

—0.14 

148.10 

ft 

8.294 

7.300 

0.120 

70 

"         "     P.M. 

40.5 

44.0 

tt 

0.6840 

0.6840   0.6840 

0.6854 

150.23 

—0.14 

150.09 

tt 

8.299 

8.200 

0.012 

71 

it         tt        tt 

36.0 

44.0 

tt 

0.9022 

0.8985   0.9003 

0.9025 

226.80 

—0.19 

226.61 

a 

8.510 

8.130 

0.045 

72 

it         tt        it 

tt 

0.9004 

0.8965   0.8984 

0.9006 

226.09 

—0.19 

225  90 

it 

8.511 

7.530 

0.115 

73 

tt         ft        tt 

34.0 

44.0 

tt 

0.9054 

0.9002  0.9028 

0.9051 

227.78 

—0.19 

227.59 

U 

8.514 

8.410 

0.012 

74 

"      11    A.M. 

22.0 

42.0 

tt 

0.9069 

0.9022   0.9045 

0.9069 

228.46 

—0.19 

228.27 

ft 

8.519 

8.330 

0.022 

75 

tt        tt       tt 

tt 

0.9071 

0.9025   0.9048 

0.9071 

228.53 

—0.19 

228.34 

ft 

8.518 

8.230 

0.034 

76 

tt        tt       ft 

tt 

0.9107 

0.9058:  0.9082 

0.9105 

229.82 

—0.19 

229.63 

fi 

8.522 

8.360 

0.019 

77 

tt        ft       tt 

28.0 

42.0 

tt 

0.9085 

0.9039   0.9062 

0.9085 

229.06 

—0.19 

228.87 

fi 

8.517 

8.460 

0.007 

78 

tt        ti        tt 

36.0 

42.0 

tt 

1.1051 

1.0929    1.0990 

1.1025 

305.98 

—0.22 

305.76 

fi 

8.707 

7.708 

0.115 

79 

tt        it        tt 

ti 

1.1057 

1.0924    1.0990 

1.1025 

305.98 

—0.22 

305.76 

it 

8.710 

8.300 

0.047 

80 

tt        tt       tf 

it 

1.1044 

1.0913    1.0978 

1.1013 

305.48 

—0.22 

305.26 

ft 

8.702 

8.400 

0035 

81 

it         ft        ti 

tt 

1.1032 

1.0910 

1.0971 

1.1006 

305.19 

—0.22 

304.97 

tf 

8.706 

8.500 

0   124 

82 

it         ft        ft 

tt 

1.1022 

1.0902 

1.0962 

1.0997 

304.82 

—0.22 

304.60 

ft 

8.708 

8.550 

0018 

83 

"         lt      P.M. 

it 

1.1020 

1.0884 

1.0952 

1.0987 

304.40 

—0.22 

304.18 

tf 

8.707 

8.600 

0.012 

84 

ti         tt        it 

46.0 

42.0 

ti 

1.0886 

1.0769 

1.0827 

1.0861 

299.19 

—0.22 

298.97 

ti 

8.693 

8.650 

0.005 

85 

ft         tt        fi 

46.0 

42.0 

ti 

1.3672 

1.3399 

1.3535 

1.3591 

418.36 

—0.35 

418.01 

It 

8.955 

8.800 

0.017 

86 

ft        ii        it 

tt 

1.3701 

1.3433 

1.3567 

1.3623 

419.83 

—0.35 

419.48 

it 

8.958 

8.550 

0.  >46 

87 

it        it        ii 

it 

1.3702 

1.3402 

1.3552 

1.3608 

419.15 

—0.35 

418.80 

ii 

8.960 

8.650 

0.035 

XX 

tt         tt        ft 

tt 

1.3678 

1.3419 

1.3548 

1.3604 

418.96 

—0.35 

418.61 

ft 

8.956 

8.900 

0.006 

89 

ft        it        it 

tt 

1.3820 

1.3505 

1.3662 

1.3719 

424.26 

—0.35 

423.91 

ti 

8.971 

8.850 

0.013 

90 

it         it        tt 

tt 

1.3794 

1.3491 

1.3642 

1.3698 

423.30 

—0.35 

422.95 

ft 

8.967 

7.950 

0.113 

91 

ft        ft        tt 

40.0 

42.0 

it 

1.3788 

1.3488 

1.3638 

1.3694 

423.11 

—0.35 

422.76 

tf 

8.962 

8.750 

0.024 

92 

"      12   A.M. 

34.0 

40.0 

ii 

1.6555 

1.5954 

1.6254 

1.6323 

550.05 

—0.41 

549.64 

it 

9.213 

9.150 

0.007 

93 

tt               tt              ft 

tt 

1.6472 

1.5888 

1.6180 

1.6249 

546.32 

—0.41 

545.91 

it 

9.210 

9.100 

0.012 

94 

tt               ii              tf 

tt 

1.6480 

1.5830 

1.6155 

1.6223 

545.02 

—0.41 

544.61 

tl 

9.207 

9.050 

0.017 

95 

tl               ft              fi 

tt 

1.6363 

1.5795 

1.6079 

1.6147 

541.21 

—0.41 

540.80 

ct 

9.201 

8.800 

0.044 

96 

ti               it              tt 

ti 

1.6519 

1.5890 

1.6204 

1.6273 

547.53 

—0.41 

547.12 

tt 

9.215 

8.900 

0.034 

97 

tf               ii              ft 

it 

1.6464 

1.5862 

1.6163 

1.6231 

545.42 

—0.41 

545.01 

ft 

9.208 

8.200 

0.109 

98 

tt               ft              tf 

36.0 

40.0 

tt 

1.6460 

1.5900 

1.6180 

1.6249 

546.32 

—0.41 

545.91 

U 

9.208 

9.000 

0.023 

99 

"         "     P.M. 

42.0 

40.0 

tt 

1.8508 

1.7655 

1.8081 

1.8144 

644.14 

—0.48 

643.66 

tt 

9.392 

9.350 

0.004 

100 

tf         tt        ii 

it 

1.8396 

1.7588 

1.7992 

1.8056 

639.48 

—0.48 

639.00 

ti 

9.383 

9.300 

0.009 

101 

tt         ft        tt 

tt 

1.8486 

1.7674  1.8080 

1.8144 

644.14 

—0.48 

643.66 

tt 

9.386 

9.100 

0.030 

102 

ft        tt        it 

43.0 

40.0 

tt 

1.8582 

1.7721 

1.8151 

1.8214   647.85 

—0.48 

647.37 

tt 

9.396 

9.000 

0.042 

103 

tt         tt        tf 

ti 

1.8687 

1.7819    1.8253 

1.8316;  653.27 

—0.48 

652.79 

ft 

9.407 

8.400 

0.107 

104 

tt         tt        it 

tt 

1.8446 

1.7565:  1.8005 

1.8069    640.17  —  0.48    639.69 

tt 

9.382 

9.250 

0.014 

105 

ft         tt        ft 

•1-2.1 

40.0       " 

1.8375    1.7538    1.7956    1.8022   637.68^  —  0.48    637.20 

it 

9.381 

9.200 

0.019 

XXI I —  CONTINUED. 

FROM  WHICH  THE  FORMULA  OF  CORRECTION  C  =0.116  (/B- 


189 


•  0.1)  IS  DETERMINED. 


Measurement. 

18 

19 

20 

21 

22 

23 

1  ft 

Difference 

between  the 

Proportion- 
al differ- 

Quanti- 
ty of 

Proportion- 
al differ- 

Remarks on  the  Force  and  Direction 

J.O 

1  / 

quantity  of 

ence,  or 

water 

ence  of  the 

of  the  Wind  at  the  Flume,  during 

Quanti- 

water pass- 

the differ- 

passing 

corrected 

the  Experiments. 

Mean 

ty  of 
water 

ing  the  Hume 

ence  in  the 

the 

quantity  as 

No. 

velocity 
of  tho 

passing 

the  mean 

column, 

deduced 

in  the 

of 

tubes 

the 

ve  ocity  of 

divided  by 

from  the 

flume  given 

tho 

through- 
out the 

flume, 
deduced 

the  tubes, 
and  the 

the  quan- 
tity de- 

mean 
velocity 

in  the 
preceding 

General  Remarks. 

flume 

from  the 

quantity 

duced  from 

of  the 

column, 

Exp. 

by  tho 

mean 
velocity 

deduce,  from 
the  weir 

the  flume 
measure- 

tubes 
correc'd 

and  the 
weir  meas- 

Force. 

Direction. 

g    m. 

of  the 

measure- 

ment. 

by  the 

urement. 

tubes. 

r\lt 

ment. 

formula 

Feet 

Cubic  ft. 

Q"  —  Q' 
Cubic  ft. 

Q" 

Q'"  = 

V 

per  sec. 

per  sec. 

per  nee. 

-0.1)  ). 

57 

1.991 

487.27 

—  4.15 

—0.0085 

488.54 

—  o.iiu.59 

Very  gentle,       1 

Down  stream. 

58 

1.969 

481.00 

—  2.31 

—0.0048 

481.00 

—0.0048 

variable.            | 

U                It 

11                 (( 

59 

1.978 

483.14 

—  0.17 

—0.0004 

478.88 

—0.0092 

it        ii 

ti         t< 

60 

2.029 

496.76 

+  4.36 

+0.0088 

490.44 

—0.0040 

Very  gentle. 

Irregular. 

- 

61 

2.013 

493.07 

—  0.21 

—0.0004 

491.33 

—0.0040 

Calm. 

62 

2.034 

498.21 

+  5.27 

+0.0106 

497.66 

+0.0096 

(  Very  gentle,       ) 
\  variable. 

Up  stream. 

63 

2.040 

498.97 

+12.47 

+0.0250 

485.65 

—0.0017 

Down  stream. 

64 
65 

1.360 
1.376 

151.08 
152.69 

—  0.33 
--  1.97 

—0.0022 
-  -0.0129 

150.42 
151.10 

—0.0065 
-  -0.0025 

|  Moderate,  but   ) 
}  variable.            ( 

Irregular. 

Reduced  copies  of  the  diagrams,  constructed 
for  experiments,  67,  68,  and  69,  are  given  in  plate 
XVII 

66 

1.385 

153.65 

--  3.33 

-  -0.021  7 

151.53 

-  -0.0080 

(C                       U 

(i 

67 

1.349 

149.69 

--  0.06 

-  -0.0004 

150.08 

-  -0.0030 

(  More  moderate,  ) 
i  but  variable.      ) 

it 

68 

1.370 

151.99 

--  3.11 

40.0205 

151.02 

-  -0.0144 

i* 

69 

1.381 

153.12 

+  5.02 

+0.0328 

148.74 

-  -0.0043 

11                       U 

" 

70 

1.373 

152.32 

+   2.23 

+0.0146 

152.15 

+0.0137 

(  Moderate,  but   ) 
{  variable.            I 

Down  stream. 

71 

2.023 

230.23 

+  3.62 

+0.0157 

227.23 

+0.0027 

72 

2.020 

229.87 

+  3.97 

+0.0173 

223.49 

—0.0107 

73 

1.990 

226.55 

_  1.04    —0.0046 

226.30 

—0.0057 

Moderate. 

Down  stream. 

74 

2.029 

231.09 

+  2.82   .+0.0122 

229.79 

+0.0067 

Calm. 

75 

2.031 

231.29 

+  2.95 

+0.0128 

229.03  +0.0030 

Very  gentle. 

Up  stream. 

76 

2.029 

231.24 

+  1.61 

+0.0070 

230.22 

+0.0026 

Moderate. 

(  Generally  up 
\  stream. 

77 

2.001 

227.94 

—  0.93 

—0.0041 

228.37 

—0.0022 

Very  gentle. 

Irregular. 

78 

2.691 

313.28 

--  7.52 

+0.0240 

304.59 

—0.0038 

Gentle. 

Down  stream. 

79 

2.663 

310.20 

--  4.44 

+0.0143 

306.00 

+0.0008 

Hardly  perceptible. 

Uncertain. 

80 

2.638 

306.91 

--  1.65 

+0.0054 

303.81 

—0.0047 

"               " 

H 

81 

2.638 

307.09 

--  2.12 

+0.0069 

305.13 

+0.0005 

It                           U 

H 

82 

2.606 

303.51 

—  1.09 

—0.0036 

302.31 

—0.0075 

t(                 (t 

u 

83 

2.607 

303.53 

—  0.65 

—0.0021 

303.19 

—0.0033 

((                           U 

U 

84 

2.526 

293.64 

—  5.33 

—0.0182 

294.64 

—0.0145 

«          « 

« 

85 

3.484 

417.20 

—  0.81 

-A).0019 

415.73 

—0.0055 

Calm. 

86 

3.534 

423.27 

+  3.79 

+0.0090 

417.65 

—0.0044 

Very  gentle. 

Irregular. 

87 

3.532 

423.21 

+  4.41 

+0.0104 

418.94 

+0.0003 

Calm. 

88 

3.454 

413.70 

—  4.91 

—0.0119 

414.78 

—0.0091 

U 

1 

89 

3.510 

421.06 

—  2.85 

—0.0068 

420.37 

—0.0084 

U 

90 

3.623 

434.38 

+11.43 

+0.0263 

422.48 

—0.0011 

U 

91 

3.531 

423.17 

+  0.41 

+0.0010 

420.47 

—0.0054 

92 

4.451 

548.35 

—  1.29 

—0.0024 

549.39 

—0.0005 

Calm. 

Reduced  copies  of  the  diagrams  constructed 

93 

4.401 

541.99 

—  3.92 

—0.0072 

541.39 

—0.0083 

Hardly  perceptible. 

Down  stream. 

for  experiments  92,  96,  and  97,  are  given  in  plate 

94 

4.415 

543.50 

—  1.11 

—0.0020 

541.59 

—  0.0055 

«                           H 

U                    U 

XVII. 

95 

4.400 

541.33 

--  0.53 

--O.OdlO 

534.44 

—0.0118 

Calm. 

96 

4.454 

548.87 

--  1.75 

-  -0.0032 

543.50 

—0.0066 

U 

97 

4.496 

553.64 

--  8.63 

-  -0.0156 

538.86 

—0.0113 

« 

98 

4.446 

547.41 

--   1.50 

-  -0.0027 

544.13 

—0.0033 

II 

99 

5.156 

647.55 

+  3.89 

+0.0060 

650.31 

+0.0103 

«( 

100 

5.079 

637.28 

—  1.72 

—0.0027 

637.66 

—0.0021 

a 

101 

5.209 

653.73 

+10.07 

-  -0.0154 

648.18 

-  -0.0070 

a 

102 

5.226 

656.67 

+  9.30 

-  -0.0142 

648.68 

-  -0.0020 

Hardly  perceptible. 

Down  stream. 

103 

5.346 

672.45 

+19.66 

-  -0.0292 

654.74 

-  -0.0030 

Calm. 

104 

5.138 

644.59 

+  4.90 

-  -0.0076 

643.22 

-  -0.0055 

" 

105 

5.136 

644.21 

+   7.01 

-  -0.0109 

641.38 

-  -0.0066 

'* 

Means  for  the 
braically 
Means  for  the  v 
signs      . 

wide  flume,  taken  algu- 

+0.0088 
0.0129 

+0.0013 
0.0080 

ide  flume,  disregarding 

Means  for  the  narrow  flume,  taken  al- 
gebraically   

+0.0072' 

... 

—0.0011 

Means  for  the  narrow  flume,  disregard- 

0.0105 

0.0057 

Taking  the  whole  105  experiments. 

Means  taken  algebraically     .       .       . 

+0.0082 

.      .      . 

+0.0004 

Means  disregarding  signs     . 

0.0120 

0.0071 

190  A  METHOD  OF   GAUGING  THE   FLOW  OF  WATER 


DESCRIPTION  OF  TABLE  XXII. 

223.  It  will  be   seen   that   the   experiments   are  divided  into   groups  of  seven. 
In   all   the    experiments   in   the   same   group,   the    quantity   of  water   passing  was  in- 
tended  to   be    the   same.     Precise   uniformity   in   the    quantity   was   not   essential   for 
the   attainment   of  the   object  in   view,  and   as  such  uniformity  would  have  required 
much  time  to  bring  about,  it  was  not  attempted.     The  width  of  the  flume  remained 
constant ;   the    depth   of  water   in   the   flume    depended   upon  the  depth  on  the  weir, 
which    was   determined   by  the    quantity  of  water   flowing,   and  which  was,  as  before 
stated,  nearly  constant.     We  have  then   in  each  group  seven  experiments,  in  which 
the   width  of    the   flume,   the    depth   of    the   water,    the    quantity   of    water    passing, 
and   the    mean    velocity   of    the    water,   are   very   nearly   constant.      The    only    ma- 
terial variation  is  in   the   length   of  the   immersed   part   of  the    tube.     For   instance, 
in    the    first    seven    experiments,    the    length    of    the    immersed    part    of  the    tube 
(column    14)   varied  from   9.482   feet    to    8.530    feet,    the    depth    of    water    in    the 
flume    (column  13)   in   the   same    experiments   remaining   nearly   constant. 

224.  Experiments  1  to  63  are    all  with  the  wide  flume,  figures  1  and  2,  plate 
XV.;    the    minute    variations    in    the    width,   given    in   column    12,   arise    from   the 
measures  having  been  taken  several   times  during  the  course ;  and  the  same  remark 
applies   to   the   length   of    the    weir,   given    in   column   4.      Experiments    64   to    105 
are   all   with   the   narrow   flume,  figures   3   and   4,   plate   XV. 

225.  Table     XXII    will    be     understood    from    the    headings     of    the     several 
columns,   together   with   what    has   been    said   previously,   without  much   further   ex- 
planation.    The    mean   observed    depth   of  water   on   the   weir  is  given  in  column  8. 
As   explained   above,  this   observed   depth   is  subject   to   several   corrections,  which  it 
has   not  been   thought   necessary  to   give    in   detail   in   the    table.     It   may  be   well, 
however,   to   indicate   them   for   one  of  the  experiments,  say  the  first,  in   which  they 
are   as   follows :  — 

Mean  observed  depth  on  the  weir, 1.8778  feet. 

Correction  for  the  difference  in  the  observed  depth,  when  the  lower 
orifice  of  the  hook  gauge  box  pipe  is  at  a  point  6  feet  from 
the  plane  of  the  weir,  instead  of  0.52  feet,  as  in  the  exper- 
iment,   .  —  0.0021  « 

1.8757    " 
Correction   for  the   velocity  of  the   water   approaching   the  weir.     See 

section   153, -)_  0.0140     " 

1.8897    « 


IN   OPEN   CANALS   OF   UNIFORM    RECTANGULAR   SECTION.  191 

Correction  for  the  obstruction  to  the  flow  over  the  weir,  by  the  apron, 

trough,   &c '    .  —  0.0058  feet. 

Corrected  depth  on  the  weir,  as  given  in  column  8 1.8839     " 

The   correction  for  the   leakage   into  the  flume    is   required   only  in  the  experi- 
ments with  the  narrow  flume,  as  is  previously  explained. 


FORMULA  OF  CORRECTION  FOR  GAUGINGS  MADE  WITH  LOADED  POLES   OR  TUBES. 

226.  The   absolute   difference   in  the  quantities  deduced  from  the  weir  measure- 
ment and  from  the  mean  velocity  of  the   tubes  is  given  in  column   18,  Table  XXII, 
and   the  proportional  difference  of  the  same  quantities  is  given  in  column   19.     The 
quantity   deduced   from   the   weir  measurement,  given   in  column  11,  is  taken  as  the 
true  quantity  passing  the  flume.     By  reference  to  columns  15  and  19  it  will  be  seen, 
that,   when   the  tube   extends   nearly   to    the    bottom    of    the   flume,    the    differences 
are   small,   generally   less   than   one    per   cent.     In    each   group   there   is    one    exper- 
iment  in   which    the    tube    does   not   extend   to   the    bottom  within    about   one   foot; 
in   these   the   differences   in  the  quantities  obtained  by  the  two  methods  are  greater, 
as   might   be    expected ;   in   these,   however,   the    differences   are,  generally,  less   than 
three  per  cent;   in  one  experiment  only  (43)  does  it  exceed  four  per  cent. 

227.  It  was  anticipated,  when  the  programme  of  the  experiments  was  arranged, 
that   the   differences   would   be    found   to   vary  with  the  velocity  of  the  water  in  the 
flume.     If  any   such   relation   exists,  it   should   be   indicated   by  the  mean   values   of 
the  proportional  differences  in  the  several  groups. 

Table   XXIII.,   arranged   according   to   velocities,   and  for    each   width    of   flume 
separately,  gives   these   mean   values. 


192 


A   METHOD   OF    GAUGING  THE   FLOW  OF   WATER 


TABLE    XXIII. 


Numbers  of  the 

Mean  Telocity  of  the 

experiments 

Width  of  the  flume  : 

the  tubes  :  in  feet  per 

Mean  proportional 

constituting  the 

in  feet. 

difference. 

second. 

group. 

43    to    49 

26.75 

0.499 

-  0.0262 

36    "    42 

(t 

1.136 

-  0.0079 

22    "    28 

U 

1.476 

-  0.0074 

29    «    35 

ft 

1.756 

-  0.0044 

15    «    21 

a 

1.983 

-  0.0024 

57    "    63 

u 

2.008 

-  0.0043 

8    "    14 

Si 

2.481 

-  0.0079 

1    "      7 

u 

2.670 

f-  0.0097 

50    "    56 

It 

2.690 

-  0.0092 

Means, 

1.855 

+  0.0088 

64    to    70 

13.37 

1.371 

-  0.0144 

71    "    77 

tl 

2.018 

-  0.0080 

78    "    84 

.     « 

2.627 

-  0.0038 

85    "    91 

II 

3.524 

-  0.0036 

92    «    98 

u 

4.438 

-  0.0016 

99    "  105 

a 

5.184 

-  0.0115 

Means, 

3.194 

+  0.0071 

Mean  proportional  difference  for  all  the  experiments. 

4-  0.0082 

228.  By   the   preceding   table   it  does  not   appear   that  the  difference    depends 
on   the   velocity.      In  both   the   wide   and   narrow   flume,   however,   the   difference   is 
greatest  when   the   velocity  is    least,   although    the   velocities   in   the   two   cases   are 
very  different.      Whether    this    is    accidental    or    depends    on    some    principle    is   a 
question   I   have   no   means   of  answering. 

229.  In   the  wide   flume,  the   mean   proportional   difference  is   0.0088,   or   about 
•J  of  one  per  cent.     In   the  narrow  flume,  the   mean  proportional  difference  is  0.0071, 
or  a  little   less   than   f   of   one  per   cent.     Thus,   on    comparing   the  whole   of   the 
experiments    in    the    two   flumes,   given    in   table   XXIII.,   it   appears   that  the   pro- 
portional differences  vary  only  0.00.17,  or  about  ^  of  one  per  cent. 

230.  The   proportional   differences   given   in   column    19    are  very  irregular,  and 
of  the   nature   of  residual    quantities,   depending   upon    errors   of  observation,  the  in- 
stability  of    the    currents   and    the   numerous   causes   tending   to   produce   differences 
in   the   results,   derived   from   the  mean  velocity  of  the  tubes  and  the  weir  measure- 
ment.     I   am   unable    to    assign   to    each    cause    its    legitimate   effect ;    all   I   can   do 
is   to   find    an    empirical    formula    that  will    represent,  with    sufficient   accuracy  for 
practical    purposes,   the   difference    in  the   usual   cases  which   occur  in   practice.     In 
arranging   the   programme  of  experiments,  it  was   designed  to   cover  the  usual  range 


IN  OPEN   CANALS   OF   UNIFORM    RECTANGULAR   SECTION.  193 

of  velocities  and  proportional  depths  of  immersion  of  the  tubes,  and  any  application 
of  the  empirical  formula  founded  on  them  will  generally  be  free  from  the  objection 
of  being  outside  the  range  of  the  experiments  on  which  it  is  founded. 

23] .  We  have  to  seek  for  an  expression  or  formula  which  will  enable  us  to 
deduce  the  real  quantity  from  that  deduced  from  the  velocity  of  the  tubes,  by 
assuming  that  they  indicate  the  mean  velocity  of  the  water  for  the  whole  depth 
of  the  part  of  the  stream  in  which  they  float. 

In  the  absence  of  experimental  data  it  would  be  rational  to  assume  that  the 
formula  of  correction  is  a  function  of  three  quantities,  viz. :  — 

1.  The   width  of  the   flume   relatively   to   its   depth. 

2.  The   mean  velocity   of  the   current. 

3.  The   depth  to   which   the   tube   is   immersed,  relatively  to   the  whole   depth 
of  the  stream. 

1.  The   sides  of  the   flume  must,  of  course,   cause  a  retardation  of  the   current 
similar   to   that   produced  by   the   bottom;   by   reference   to   the  several  diagrams  on 
plate  XVII.   it  will  be  seen   that   the  velocity   of  the   tubes   is  diminished  near  the 
sides.     It   is  not  practicable   to  measure    the  velocity,  by  means  of  the    tubes,  quite 
close   to   the   sides,  but   in   drawing   the   curves,  representing  the  mean  velocities  of 
the   tubes,  it  will   be    seen   that    the    retarding   effects   of   the   sides   are   attempted 
to   be   allowed   for. 

We  have  experiments  only  on  flumes  of  two  widths,  one  being  twice  the 
width  of  the  other;  the  depths  being  nearly  the  same,  the  relative  width  in  one 
will  be  about  twice  that  in  the  other.  By  reference  to  table  XXIII.  it  will  be 
seen  that  in  the  wide  flume  the  mean  proportional  difference  is  -|-  0.0088,  the 
mean  velocity  being  1.855  feet  per  second.  In  the  narrow  flume,  if  we  take  the 
whole  of  the  experiments,  the  mean  velocity  is  much  greater  than  in  the  exper- 
iments in  the  wide  flume.  If,  however,  we  take  the  three  first  groups,  which  in- 
clude experiments  No.  64  to  84,  we  have  for  the  mean  velocity  2.005  feet  per 
second,  and  a  mean  proportional  difference  of  -j-  0.0087.  Comparing  the  results 
from  the  two  flumes,  it  appears  that  by  doubling  the  relative  width,  other  cir- 
cumstances remaining  nearly  the  same,  the  proportional  difference  has  not  been 
sensibly  affected.  We  may,  therefore,  conclude  that  the  relative  width  of  the  flume 
need  not  enter  into  the  formula  of  correction,  care  being  taken,  in  drawing  the 
curves,  representing  the  mean  velocities  in  different  parts  of  the  width  of  the  flume, 
to  inflect  the  curve  downwards  at  the  sides,  as  has  been  done  in  reducing  these 
experiments. 

2.  As  depending  on  the  mean  velocity  of  the  current.     It  results,  from  Navier's 

25 


194  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

investigation,  that,  so  far  as  it  depends  on  the  excess  of  the  velocity  of  the  tube 
above  that  of  the  water  in  which  it  is  floating,  the  absolute  difference  is  pro- 
portional to  the  velocity  (art.  196);  the  proportional  difference,  which  we  are  con- 
sidering, must  therefore  be  constant,  or  independent  of  the  velocity.  By  reference 
to  table  XXIII.  it  will  be  seen  that  the  mean  proportional  differences  in  the 
several  groups  of  experiments  in  each  flume  appear  to  have  two  maxima  and  one 
minimum ;  the  experiments  in  which  the  velocities  are  least  and  greatest  having 
the  greatest  proportional  difference,  and  some  intermediate  velocity  having  the  least 
proportional  difference.  Comparing  the  whole  of  the  experiments  in  both  flumes, 
we  find  in  the  group  having  the  least  velocity  the  largest  proportional  difference ; 
but  this  result  having,  apparently,  no  connection  with  the  results  deduced  from  the 
great  mass  of  the  experiments,  must,  until  explained,  be  considered  anomalous. 
Comparing  the  results  deduced  from  all  the  experiments,  excepting  those  comprised 
in  the  first  group,  no  connection  can  be  traced  between  the  velocities  and  the 
mean  proportional  differences.  We  must  therefore  conclude,  that  the  correction  is 
independent  of  the  velocity. 

3.  As  depending  on  the  depth  to  which  the  tube  is  immersed,  relatively  to 
the  whole  depth  of  the  stream.  It  is  evident  that,  in  the  cases  in  which  the  nat- 
ural scale  of  velocities  at  different  depths  has  become  established,  the  difference  in 
question  must  depend  mainly  upon  this  circumstance,  and  its  magnitude  may  be 
computed  by  the  formulas  of  Humphrey  and  Abbot  together  with  those  of  Navier 
and  Frizell,  as  has  been  previously  shown  (arts.  193,  196);  but  in  these  experiments, 
and  in  the  cases  which  usually  occur  in  practice,  this  natural  relation  is  not  estab- 
lished, and  consequently  these  formulas  do  not  apply;  and  there  appears  to  be  no 
alternative  but  to  determine  an  empirical  formula  from  the  experiments,  which  will 
serve  for  practical  purposes. 

232.  In  determining  the  formula  of  correction,  it  is  assumed  that  the  pro- 
portional difference  depends  only  upon  the  relative  depth  to  which  the  tube  is 
immersed.  Instead  of  using  this  relative  depth,  it  has  been  found  more  convenient 
to  use  a  quantity  depending  directly  upon  it,  viz.  the  difference  between  the 
depth  of  the  water  in  the  flume  and  the  depth  to  which  the  tube  is  immersed, 
divided  by  the  depth  of  the  water  in  the  flume;  this  we  call  D,  and  its  value  in 
each  experiment  in  table  XXII.  is  given  in  column  15. 

For  the  purpose  of  more  convenient  graphic  representation,  the  data  given  in 
table  XXII.  are  reduced,  by  taking  means  of  the  values  of  D  within  certain 
limits,  and  also  of  the  corresponding  values  of  the  proportional  differences  ^"~  • 
given  in  column  19.  These  means,  arranged  according  to  the  values  of  D,  are 
given  separately  for  each  width  of  flume,  in  table  XXIV. 


IN   OPEN    CANALS   OF    UNIFORM   RECTANGULAR   SECTION. 


195 


TABLE    XXIV. 


Number  of 

Greatest  and  least  values  of  D 

Me 

in  value  of  the 

Width  of  the 

experiments 

in  the  experiments  from  which 

proportional  difference. 

flume. 

from  which  the 

the  means  are  deduced. 

Mean  value  of  D. 

means  are 

Q»-Q< 

Feet. 

deduced. 

Greatest. 

Least. 

Q" 

26.746 

9 

0.007 

0.004 

0.0054 

+  0.00129 

(( 

12 

0.012 

0.008 

0.0107 

-  0.00027 

ff 

8 

0.017 

0.015 

0.0165 

-  0.00400 

H 

7 

0.023 

0.019 

0.0211 

-  0.00251 

u 

9 

0.034 

0.029 

0.0318 

-  0.00856 

U 

9 

0.055 

0.041 

0.0446 

-  0.01577 

It 

9 

0.129 

0.104 

0.1118 

+  0.03033 

13.372 

6 

0.007 

0.004 

0.0058 

—  0.00503 

tt 

5 

0.012 

0.009 

0.0114 

—  0.00040 

u 

4 

0.017 

0.013 

0.0152 

—  0.00080 

u 

9 

0.024 

0.018 

0.0213 

+  0.00616 

u 

5 

0.035 

0.030 

0.0336 

-j-  0.00944 

tt 

7 

0.048 

0.036 

0.0440 

+  0.01269 

tt 

6 

0.120 

0.107 

0.1132 

4-  0.02420 

233.     In   the   diagram  figure   2,   plate   XVIII.  the  abscissas   represent   the  mean 
values    of    D    in    the    preceding    table    and    the   ordinates   the  corresponding   mean 

Qn  Qi 

values  of  the  proportional  differences  -'  Q,,  ,  the  double  circles  representing  the 
experiments  with  the  wide  flume,  and  the  single  circles  the  experiments  with  the 
narrow  flume.  As  will  be  seen,  the  parabolic  curve  A  B  represents,  nearly, 
the  mean  result  of  all  the  experiments.  Calling  the  ordinates  of  jthe  curve  (7,  and 
the  abscissas  D,  its  equation  is 


v/Z>  —  0.1)  (1.) 

C  is  the  proportional  difference  to  be  deducted  from  the  quantity  directly 
deduced  from  the  mean  velocity  of  the  tubes;  Q"  being  the  quantity  thus  deduced 
and  Q'"  being  the  corrected  quantity,  we  have 


substituting  the  value  of  C  in  (1.),  we  have 

Q'"  =.(!_.  0.116  (yTD  —  0.1))  Q". 
Table  XXIX.  gives  the  values  of  the  coefficient 

1  —  0.116  (y/~^  —  0.1) 


(2.) 


196  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

for   the    values    of  D   from    0.000    to   0.100,   together   with   the    logarithms    of  the 
same. 

234.  Column  20,  table  XXII,  gives  the  values  of  Q"f  t>y  formula  (2.),  and 
column  21  the  proportional  differences  between  the  values  of  Q'"  and  the  quantities 
as  measured  at  the  weir.  Taking  the  whole  of  the  experiments  together,  it  will  be 
seen  that  the  mean  proportional  difference,  taken  algebraically,  is  -f-  0.0004,  or,  dis- 
regarding the  signs,  0.0071;  the  latter  quantity  is  about  f  of  one  per  cent,  and  is 
the  mean  error  or  discrepancy  between  the  measurement  by  the  weir  and  the  cor- 
rected measurement  in  the  flume.  It  will  be  observed  that  the  largest  discrepancies 
are  in  the  group  of  experiments  numbered  from  43  to  49,  in  which  the  velocity 
was  very  slow;  in  one  of  these  experiments,  viz.  No.  46,  the  corrected  flume 
measurement  is  about  -^  greater  than  the  weir  measurement.  In  experiment  No. 
47  the  corrected  flume  measurement  is  about  -fa  greater  than  the  weir  measurement. 
In  experiment  No.  16  the  corrected  flume  measurement  is  about  TJ¥  less  than  the 
weir  measurement.  In  all  the  other  experiments,  the  difference  is  less  than  -fa, 
or  two  per  cent 


198 


TABLE 
MISCELLANEOUS  EXPERIMENTS  AT  THE 


1 

3 

3 

Weir  Measurement. 

Flume 

Tempera- 

4 

5 

O 

7 

8 

9 

10 

11 

ia 

13 

14 

15 

degrees  of 
Fahrenheit's 

Quant  it; 

of  wate 

Corrects 
quantit 

between 
the  depth 

Date. 

thermometer 

Total 

Observe* 

Observec 

Mean 

Corrects 

passing 
over  the 

Correc- 

passing 

r  helium 

Mean 

Length 

of  water 
in  the 

No. 
of 

length 
of  the 
weirs. 

depth  o: 
water  on 
the 

depth  of 
water  on 
the 

observed 
depth  of 
water 

depth  ol 
water 
on  the 

weirs, 
com- 
puted 

tion  for 
the  leak- 
age into 

deduce* 
from  the 
weir 

Mean 

width  ol 
the  flume 

depth  o 
water 
in  the 

r  of  the 
im- 
mersed 

flume  an< 
the  length 
of  the 

the 

1856. 

of  the 

Westerly 
weir. 

Easterly 
weir. 

on  the 
weirs. 

weirs. 

by  the 
formula 

the  flume 

measure 

1  1  iciit. 

flume. 

part 
of  the 

immersed 
part  of 

phere 
in 
shade. 

of  th 
water 

L 

H 

3.31 

Q  = 

!(£-0.08n/ 

OH*' 

Q' 

tube. 

the  tube, 
divided  by 
the  deptt 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Cubic  ft. 

Cubic  ft. 

Cubic  ft 

Feet. 

Feet. 

Feet. 

of  water  in 
the  flume. 

per  sec. 

per  sec. 

per  sec. 

D 

106 

Oct.  22  A.M. 

53.0 

53.0 

80.014 

0.5509 

0.5508 

0.5508 

0.5518 

108.58 

0.00 

108.58 

26.745 

8.177 

8.070 

0.013 

107 

it             it             ti 

57.5 

53.0 

u 

0.5501 

0.5501 

0.5501 

0.5511 

108.38 

0.00 

108.38 

tt 

8.173 

8.070 

0.013 

108 

it             11            It 

60.5 

53.0 

ft 

0.5504 

0.5505 

0.5504 

0.5514 

108.46 

0.00 

108.46 

tt 

8.175 

8.070 

0.013 

109 
110 
111 

NOV.  13  A.M. 

ti          if          it 
U          ((          it 

32.0 

40.0 

80.012 
(t 

u 

1.1583 
1.1661 
1.1634 

1.1358 
1.1435 
1.1409 

1.1470 
1.1548 
1.1521 

1.1508 
1.1587 
1.1560 

326.23 
329.59 
328.44 

—0.27 
—0.27 

—0.27 

325.96 
329.32 

328.17 

13.372 

ft 
tt 

8.760 
8.766 
8.765 

8.650 
8.650 
8.650 

0.013 
0.013 
0.013 

112 

(t           ff          U 

36.0 

40.0 

H 

1.1604 

1.1373 

1.1488 

1.1527 

327.04 

—0.27 

326.77 

tt 

8.760 

8.650 

0.013 

113 

Oct.  23  P.M. 

50.5 

53.0 

80.014 

1.8580 

1.8362 

1.8471 

1.8534 

664.91 

0.00 

664.91 

26.745 

9.529 

9.430 

0.010 

114 

"      30   A.M. 

tt 

1.3612 

1.3509 

1.3560 

1.3616 

419.51 

0.00 

419.51 

26.746 

9.003 

8.600 

0.045 

115 

it        it       ti 

67.0 

47.0 

tt 

1.3685 

1.3588 

1.3636 

1.3692 

423.02 

0.00 

423.02 

tt 

9.016 

8.900 

0.013 

116 

"        "     P.M. 

67.0 

47.0 

ti 

1.3391 

1.3350 

1.3370 

1.3424 

410.70 

0.00 

410.70 

ft 

9.015 

8.000 

0.113 

117 

it          U         it 

66.0 

47.0 

tt 

1.3291 

1.3255 

1.3273 

1.3327 

406.28 

0.00 

406.28 

(f 

9.010 

8.850 

0.018 

118 

tt       it       it 

64.0 

47.0 

tt 

1.2405 

1.2221 

1.2313 

1.2358 

362.92 

0.00 

362.92 

ff 

8.882 

7.852 

0.116 

119 

"     31      « 

0.9921 

0.9857 

0.9889 

0.9917 

261.15 

0.00 

261.15 

ft 

8.628 

8.220 

0.047 

120 
121 

Oct.  22  P.M. 
"     23     " 

54.0 
48.0 

53.0 
53.0 

80.014 

it 

1.8994 
1.8408 

1.8539 
1.8144 

1.8766 
1.8276 

1.8826 
1.8339 

680.61 
654.50 

0.00 
0.00 

680.61 
654.50 

26.745 

tt 

9.526 

9.500 

9.429 
9.430 

0.010 
0.007 

122 

it      ti      tt 

45.5 

53.0 

ti 

1.8244 

1.8001 

1.8122 

1.8186 

646.36 

0.00 

646.36 

tt 

9.479 

9.430 

0.005 

123 
124 

Nov.  4  A.M. 

ti       (I       ti 

52.0 
53.0 

47.0 
47.0 

80.014 
tt 

1.8837 
1.8682 

1.8450 
1.8321 

.8643 
.8501 

1.8704 
1.8563 

674.03 
666.47 

0.00 
0.00 

674.03 
666.47 

26.746 

ft 

9.527 
9.518 

9.430 
9.430 

0.010 
0.009 

125 

tt       tt       tt 

53.0 

47.0 

tt 

1.8588 

.8230 

.8409 

1.8472 

661.60 

0.00 

661.60 

"  tt 

9.509 

9.431 

0.008 

126 

(t       if       It 

54.0 

47.0 

tt 

1.8550 

.8193 

.8371 

1.8433 

659.51 

0.00 

659.51 

it 

9.506 

9.432 

0.008 

127 

tt       tt       {{ 

54.0 

47.0 

tt 

1.8378 

.8023 

.8200 

.8263 

650.46 

0.00 

650.46 

ft 

9.490 

9.434 

0.006 

128 

((      tt      tt 

56.0 

47.0 

u 

1.8217 

.7888 

.8052     .8116 

642.65 

0.00 

642.65 

tt 

9.476 

9.436 

0.004 

129 

tt       tt       (t 

56.0 

47.0 

t 

1.8692 

.8328 

.8510     .8572 

666.95 

0.00 

666.95 

ft 

9.529 

9.437 

0.010 

130 

"        "     P.M. 

56.0 

47.0 

t 

1.8731 

.8271 

.8501      .8563 

666.47 

0.00 

666.47 

tt 

9.555 

9.439 

0.012 

131 

tt        tt       tt 

58.0 

47.0 

t 

1.8670 

1.8233 

.8451      .8514 

663.85 

0.00 

663.85 

ft 

9.551 

9.445 

0.011 

132 

tt        tt       ti 

58.0 

47.0 

« 

1.8603 

1.8166 

.8384 

.8447 

660.26 

0.00 

660.26 

tt 

9.549 

9.440 

0.011 

133 

it        tt        ti 

58.0 

47.0 

t 

1.8376 

1.7950 

1.8163 

1.8226 

648.49 

0.00 

648.49 

tt 

9.529 

9.430 

0.010 

134 

ft        ft       it 

58.5 

47.0 

t 

1.8780 

1.8435 

1.8607 

1.8669 

672.16 

0.00 

672.16 

tt 

9.562 

9.430 

0.014 

135 
136 
137 
138 
139 
140 

Nov.  5  A.M. 

tt       tt       tt 
tt      ff      ft 
tl       ((       ft 
ft       ft       ft 
tt       tt       ff 

39.0 
40.0 
40.0 
41.0 

48.0 
48.0 
48.0 
48.0 

80.014 
tt 
tt 
tt 
tt 
tt 

1.4879 
1.4820 
1.4823 
1.4832 
1.4855 
1.4794 

1.4695 
1.4631 
1.4659 
1.4679 
1.4724 
1.4691 

1.4787 
1.4725 
1.4741 
1.4755 
1.4789 
1.4740 

1.4852 
1.4790 
1.4806 
1.4820 
1.4854 
1.4805 

477.68 
474.70 
475.46 
476.14 
477.77 
475.42 

0.00 
0.00 
0.00 

o.oo  i 

0.00 
0.00 

477.68 
474.70 
475.46 
476.14 
477.77 
475.42 

26.746 

tt 
tt 
tt 
tt 
tt 

9.164 
9.159 
9.168 
9.175 
9.176 
9.174 

9.040 
9.040 
9.040 
9.040 
9.040 
9.040 

0.014 
0.013 
0.014 
0.015 
0.015 
0.015 

XXV. 

TREMONT  WEIR  AND  MEASURING  FLUME. 


199 


Measurement. 

18 

19 

ao 

SI 

23 

33 

1  f\ 

1  "Y 

Difference 
between  the 

Proportion- 
al differ- 

Quanti- 
ty of 

Proportion- 
al differ- 

Remarks on  the  Force  and  Direction 

A.\J 

1    1 

quantity  of 

ence,  or 

water 

ence  of  the 

of  the  Wind  at  the  Flume,  during 

Quanti- 

water pass- 

the dilTer- 

passing 

corrected 

the  Experiments. 

Mean 

ty  of 
water 

ing  the  flume  ence  in  the 

the 

quantity  as 

No. 

velocity 
of  the 

passing 

the  mean 

column, 

deduced 

in  the 

of 

tubes 

the 

velocity  of 

divided  by 

from  the 

flume  given 

the 

through- 
out the 

flume, 
deduced 

the  tubes, 
and  the 

the  quan- 
tity de- 

mean 
velocity 

in  the 
preceding 

General  Remarks. 

flume 

from  the 

quantity 

duced  from 

of  the 

column, 

Exp. 

by  the 

mean 
velocity 

deduced  from 
the  weir 

the  flume 
measure- 

tubes, 
correct 

and  the 

weir  meas- 

Force. 

Direction. 

e>      Ul. 

of  the 

measure- 

ment. 

by  the 

urement. 

tubes. 
f\n 

ment. 

Q"-Q' 

formula 

Q"'-Qr 

Feet 

V 
Cubic  ft. 

Cubic  ft. 

~Q"   ~ 

Q'"  = 
U.H6(v'l> 

Q'   . 

-0.1)). 

per  sec. 

per  sec. 

per  sec. 

*  ^ 

106 
107 
108 

0.503 

0.502 
0.504 

110.09 
109.62 
110.10 

+  1.51 
+  1.24 
+  1.64 

+0.0137 
+0.0113 
+0.0149 

109.91 
109.44 
109.92 

+(P.  Ill  •>•> 

+0.0098 
+0.0135 

Calm. 

u 
u 

1 

These  three  experiments  were  mode  under  circum- 
stances as  nearly  identical  OH  practicable,  for  the  pur- 
pose of  testing  the  degree  of  uniformity  attained  in 
the  results.    The  greatest  variation  in  the  proportional 
differences  in  column  21,  from  the  mean,  is  0.0020, 

or  TJoff* 
Reduced  copies   of  the    diagrams   constructed    for 

these  experiments  are  given  on  plate  XVII. 

109 

•2.7.".!' 

322.34 

—  3.62 

—0.0112 

321.81 

—0.0127 

Calm. 

These  four  experiments  were  also  made  under  cir- 
cumstances as  nearly  identical  as  practicable,  for  the 

110 
111 

2.811 
2.820 

329.48 
330.49 

+  0.16 
+  2.32 

+0.0005 
+0.0070 

328.94 
329.95 

—0.0012 
+0.0054 

Very  gentle. 

It              It 

Down  stream. 

(I                 (1 

same  purpose  as  tho  preceding.    The  greatest  variation 
in  the  proportional  differences  in  column  21,  from  the 
mean,  is  0.0107  =  -gV- 

112 

2.796 

327.54 

+  0.77 

+0.0024 

327.01 

+0.0007 

Calm. 

Reduced  copies  of  the    diagrams  constructed    for 

these  experiments  we  given  on  plate  XVII. 

113 

2.542 

647.88 

—17.03 

—0.0263 

647.88 

—0.0256 

I  Very  strong,      » 
j  variable.            J 

(  Irreg.,  gen. 
|  down  stream. 

These  seven  experiments  were  made  when  the  wind 
was  blowing  with  considerable  force  in  the  direction 

114 
115 

.771 
.725 

426.55 
416.02 

+   7.04 
—   7.00 

+0.0165 
—0.0168 

421.00 
415.34 

+0.0036 
—0.0182 

Brisk. 

u 

Down  stream. 

u           u 

of  the  current.    It  will  he  seen  that  the  results  are  less 
regular  than  in  the  preceding  seven  experiments,  and 
in  the  experiments  in  Table  XXII.,  in  none  of  which 

116 

.759 

424.07 

--13.37 

+0.0315 

112.45 

+0.0043 

u 

u            u 

was  there  much  wind.    The  menu  proportional  differ- 
ence in  column  21,  in  these  seven  experiments,  is 

117 

.695 

408.53 

--  2.25 

+0.0055 

406.91 

+0.0016 

Strong. 

*(         (1 

—  O.MV.M,  which  would  indicate  that  the  wind  blowing 
down  stream  had  a  small  effect  in  diminishing  the  ve- 

118 

.563 

371.37 

+  8.45 

+0.0228 

361.01 

—0.0053 

H 

((                 U 

locity  of  the  tubes,  hut  the  irregularities  are  too  great 

119 

.174 

270.84 

--  9.69 

+0.0358 

267.17 

+0.0231 

(  Brisk,  strong     ) 

U                U 

to  permit  this  inference  to  he  drawn.    All  that  can  be 
salelv  inferred  is,  that  the  wind  had  no  sensible  effect 

(  at  times.            ) 

on  the  velocity  of  the  tubes,  except  to  increase  the 
Irregularities. 

120 
121 
122 

2.720 
2.494 
2.531 

693.06 
633.68 
641.63 

+12.45 
—20.82 
—  4.73 

+0.0180 
—0.0329 
—0.0074 

693.06 
634.88 
643.81 

+0.0183 
—0.0300 
—0.0039 

Very  moderate. 
i  Very  strong,       j 
(  variable.              J 
«         it 

Up  stream. 
(  Irreg.,  gen. 
'(  up  stream. 

In   these  three  experiments  the  wind  was  blowing 
generally   in  the  opposite    direction  to  the  current. 
The  mean  proportional  difference  in  column   21  is 
—  0.0*132,  hut  the  results  are  too  irregular  to  permit  any 
inference  to  be  drawn,  except  that  the  effect  of  the 

wind  was  insensible,  except  in  increasing  the  irregu- 
larities. 

123 

2.595 

661.35 

—12.68 

—0.0192 

661.35 

—0.0188 

Calm. 

These  twelve  experiments  were  miule  with  an  ob- 
struction placed  in  the  canal  about  150  feet  above  the 

124 
125 

2.611 
2.599 

664.76 
661.00 

—  1.71 
—  0.60 

—0.0026 
—0.0009 

665.16 
661.81 

—0.0020 
+0.0003 

Hardly  perceptible. 
Calm. 

Up  stream. 

flume  (K  F,  plate  XV.  and  art.  201),whtch  caused  great 
disturbances  in  the  motion  of  the  water  in  the  flume, 
every  part  of  it  betiig  tilled  with  eddies,  both  horizon- 

126 

2.594 

659.61 

+  0.10 

+0.0002 

660.41 

+0.0014 

u 

tal  and  vertical.    The  mcun  proportional  difference  in 
column  21,  disregarding  the  signs,  is  0.0)21.    In  table 

127 

128 

2.613 

2.488 

663.27 
630.54 

+12.81 
—  12.11 

+0.0193 
—0.0192 

665.00 
633.23 

+0.0224 
—0.0147 

u 

XXII,  the  corresponding  mean  is  0.0071.    Hence  we 
infer,  that  the  irregularities  were  greater  when  the  cur- 
rent was  disturbed  in  the  manner  described  than  when 

129 

2.539 

647.09 

—19.86 

—0.0307 

647.09 

—0.0298 

Very  gentle. 

Up  stream. 

undisturbed,  in  the  ratio  of  17  to  10.    The  mean  pro- 
portional difference  in  column  21,  regarding  the  signs, 

130 

2.633 

672.98 

+  6.51 

+0.0097 

672.23 

+0.0086 

K        a 

(t        *i 

is  —  0.0021,  from  which,  considering  the  irregularities, 
all  that  can  be  inferred  is,  that  the  disturbance  of  the 

131 

2.628 

671.36 

+  7.51 

+0.0112 

670.98 

+0.0107 

Hardly  perceptible. 

if          u 

current  has  no  sensible  effect  on  the  results,  except  in 
increasing  the  irregularities. 

132 

2.566 

655.29 

—  4.97 

—0.0076 

654.92 

—0.0081 

Calm. 

In  experiments  123,  124,  127,  128,  132,  and  133,  the 

133 

2.515 

640.98 

—  7.51 

—0.0117 

640.98 

—0.0116 

u 

tubes  were  put  into  the  water  in  regular  order,  from 
the  left  to  the  right  side  of  the  flume,  at  intervals  of 

134 

2.677 

684.64 

+12.48 

+0.0182 

683.18 

+0.0164 

" 

about  one  foot,  passing  once  across  in  the  usual  man- 
ner.   In  experiments  125,  126,  l^y,  130,  13!,  and  134, 
they  were  put  in  in  the  same  order,  but  at  .intervals  of 

about  four  feet,  and  passing  across  the  flume  ibur  times, 

in   each  experiment,  taking  different  points  at  each 

crossing.    It  was  thought  possible  that  the  positions  of 

the  quick  and  slow  currents  might  not  remain  constant 

throughout  an  experiment,  which,  with  the  ordinary 

mode  of  putting  in  the  tubes,  might  lead  to  an  errone- 

ous result.    Comparing  the  results  obtained  by  the  two 

methods,  and  disregarding  the  signs,  tbe  mean  propor- 

tional difference  in  column  21,  in  the  six  experiments 

in  which  the  tubes  were  put  in  in  the  usual  manner,  is 

I 

0.012!*.     In  the  other  six  experiments  the  mean  propor- 

tional difference  is  0.0112.    The  small  difference  in  the 

results,considering  the  irregularities,  cannot  be  attrib- 

uted to  the  mode  of  putting  in  the  tubes. 

Reduced  copies  of  the  diagrams  constructed  for  ex- 

periments 123  and  124  are  given  in  plate  XVII. 

135 
136 

2.045 
1.947 

501.32 

477.06 

+23.64 
+  2.36 

+0.0472 
+0.0049 

500.25 
476.28 

-  -0.0472 
-  -0.0033 

Strong. 

Down  stream. 

U                (t 

These  six  experiments  were  made  under  similar  cir- 
cumstances to  the  twelve  experiments  next  preceding) 
except  that  there  was  a  high  wind  blowing  in  the  di- 
rection of  the  current.    The  mean  proportional  differ- 

137 
138 
139 

2.032 
1.960 
1.923 

498.21 
480.85 
471.86 

+22.75 
+  4.71 
—  5.91 

+0.0457 
+0.0098 
—0.0125 

497.15 
479.59 
470.63 

-  -0.0456 

-  -0.0072 
—0.0149 

(  Strong,  but        I 
j  variable.             J 

" 

(t                U 

ence  in  column  21  is  +0.0207,  which  indicates  that  the 
joint  effect  of  the  disturbance  of  the  current,  and  the 
strong  wind  blowing  in  the  direction  of  the  current, 
was  to  increase  the  velocity  of  the  tubes  about  two  per 
cent. 

140 

2.012 

493.77 

+18.35 

+0.0372 

492.48 

+0.0359 

Violent. 

"      " 

In  experiments  135,  136,  130,  and  140  the  tubes  were 
put  into  the  water  in  tho  usual  manner.    In  experi- 

ments l->7  and  l-'X  they  were  put  in  as  described  above 

in  experiments  125,  126,  Ac.    The  mean  proportional 

difference  in  column  21,  in  the  four  experiments  in 

which  the  tubes  were  put  in  in  the  usual  manner,  dis- 

regarding the  signs,  is  0.0203,  and  in  the  other  two  ex- 

periments the  mean  is  0.0264  ;  indicating  no  sensible 

difference  in  the  magnitude  of  the  irregularities,  de- 

pending on  the  manner  in  which  the  tubes  were  put 
into  the  water. 

Reduced  copies  of  the  diagrams  constructed  for  ex- 
periments 138,  139,  and  140  are  given  in  plate  XVII. 

200  A  METHOD  OF   GAUGING  THE  FLOW  OF  WATER 


DESCRIPTION  OF  TABLE  XXV. 

235.  The  experiments  in  this  table  were  made   like   those   in   table   XXII.,  and 
have   been   reduced  in  the  same  manner.     The  special  purposes   for  which  they  were 
made   are   described   in   the   final   column   of    the    table,   headed   "General   Remarks." 
By    referring   to    the   table,   it   will   be    seen    that   the   first   seven  experiments   were 
made   for   the   purpose    of  testing   the  degree  of  uniformity  attainable  in  the  results, 
when   the   circumstances    under   which   the   measurements  were   made  were  the  same. 
This   is   a   fundamental   question   in   all   kinds   and  methods  of  measuring,  and  is  dis- 
tinct from   the   errors   of  observation   to   which   all   methods   are  liable.     In  geodesic 
and   astronomical    methods   the    difficulties   arise    principally   from    the   instability   of 
instruments   and   from   atmospheric   changes.      In   measuring   the   velocity   of  streams 
of   water,    the    instability   of    the    currents,   mentioned    in   article    208,    appeared    to 
afford  a  peculiar  liability  to  this  trouble,  and  it  was  necessary  to  make  special  exper- 
iments  to  ascertain   the   magnitude    of  the    irregularities    due    to    it.     In   the    three 
experiments,   numbered    106   to  108,  in  which  the  circumstances  were  as  nearly  alike 
as  practicable,  the  extreme  variation  is  about  ^{7 ;   in  the  next  group  of  four  exper- 
iments,  in    which    the    circumstances   were    also   alike,   as    nearly   as    practicable,   the 
extreme  variation  is  about  -fa ;  so  far  as  is  known,  there  was  no  want  of  care  in  the 
execution   of  any  of  these    experiments,  and  the    irregularities  must  be  considered  as 
inseparable   from   the    method.      In  a  greater  number   of  trials  the  extreme  variation 
would   probably  be  greater.     We   must   infer   from  these  seven  experiments,  that  any 
single    measurement   is    liable    to    be    erroneous   to    the   amount   of  one    per  cent,  or 
perhaps   rather   more ;   and    in   any  two   experiments   the    errors   may  be   in  opposite 
directions,   in   which   case   they  may  vary  from   each   other   two    per   cent,   or    rather 
more.      It    is    of  course    very    desirable    that    the   method    should    be    free   from  this 
liability    to    error;    except   by    accident,    however,    the    quantity    of  water  used    at   a 
manufacturing   establishment   or   flowing   in   a   stream  will    not  be  found    twice  alike. 
An   approximation   within   one    or   two  per  cent   of  the   truth    is   sufficient    for   most 
practical   purposes;   the    errors   are   as   liable  to  be  one  way  as  the  other,  and  by  re- 
peating  the   measurement   several   times   and   taking   the  mean,  the  probabilities   are 
that   the  result  will   be   very  nearly  as   correct  as   if  the  method  was  free  from  this 
liability  to  error  in  a  single  measurement. 

236.  The   seven   experiments   numbered  from  113  to  119  were  made,  when  the 
wind  was  blowing  with  considerable  force  down  stream.      Taking  the  mean,  it    would 
appear  that  the  effect   of  the  wind  was  to    cause  the    corrected   flume   measurements 
to  be  about  one  quarter  of  one  per  cent  less  than  the  weir  measurements.     In  these 


IN  OPEN   CANALS   OF   UNIFORM   RECTANGULAR   SECTION.  201 

experiments  the  length  of  the  immersed  parts  of  the  tubes  varied  from  7.85  feet  to 
9.43  feet ;  the  length  projecting  above  the  water,  in  each  case,  was  about  0.33  feet ; 
taking  a  mean,  about  ^  part  of  the  length  projected  above  the  surface  of  the 
water,  and  was  liable  to  be  acted  upon  by  the  wind.  The  effect  of  the  wind 
blowing  down  stream,  with  a  velocity  greater  than  that  of  the  current,  must  be  to 
give  the  tube  a  greater  velocity  than  it  would  have  in  a  calm  or  with  the  wind 
blowing  up  stream.  By  the  mean  result  of  the  seven  experiments  the  contrary  effect 
would  appear  to  have  been  produced.  By  comparing  the  differences  in  these  seven 
experiments,  given  in  column  21,  with  the  corresponding  differences  in  table  XXII., 
it  will  be  seen  that  the  irregularities  in  the  results  of  the  measurements  were 
much  greater  when  the  wind  was  blowing  strongly  than  when  it  was  calm,  or 
nearly  so.  The  extreme  variation  in  the  seven  experiments  is  nearly  five  per  cent; 
under  these  circumstances,  it  is  apparent  that,  in  order  to  detect  with  certainty  so 
small  a  difference  as  one  quarter  of  one  per  cent,  a  much  larger  number  of  exper- 
iments is  necessary,  and  that,  with  the  small  number  made,  the  real  effects  may 
easily  be  obscured  by  the  irregularities. 

237.  In   experiments   121   and    122    the   wind   was    very  strong,    but  variable, 
irregular  in  direction,  but  generally  up  stream;   the  mean  result  of  the   two  exper- 
iments is,  that  the  velocity  of  the  tubes  was  retarded   about  g1^ ;   but  the  number  of 
experiments  is  evidently  insufficient  to   determine  it  definitely.      We  may  infer  from 
the  ten  experiments,  numbered  from  113  to  122,   that,  although  measurements  made 
when  the  wind  is  blowing  strongly,  either  up  stream  or  down  stream,  are  subject  to 
greater  irregularities   than   measurements  made  when   there  is  little   or  no  wind,   by 
making  a  considerable  number  of  trials,  the  mean  results  will   vary  but  little,  whether 
the  wind  is  blowing  strongly  or  not. 

238.  In  the  twelve  experiments,  numbered  123  to   134,  there  was  a  great  com- 
motion in  the  stream  caused  by  an  obstruction  in  the  channel  above,  as  is  explained 
in   the   table.     The   irregularities   are   increased,  but   the  mean  result  is  not   sensibly 
affected.      In  the  six  experiments  numbered  135  to  140  there  was  a  similar  agitation 
in  the  stream,  and  also  a  high  wind  blowing  down  stream ;   the  effect  was  to  increase 
the   irregularities  in  the  results,  and  the  mean  velocity  of  the  tubes  appears  to  have 
been  increased  about  two  per  cent. 


APPLICATION  OF  THE  METHOD  OF  GAUGING  STREAMS  OF  WATER  BY  MEANS  OF 

LOADED  POLES  OR  TUBES. 

239.     As  previously  stated,  this  method  is  more  generally  adopted  at  Lowell,  for 
gauging  large   volumes  of  water,  than  any  other.     Six   measuring  flumes   have  been 
26 


201}  A  METHOD  OF    GAUGING   THE   FLOW   OF   WATER 

constructed  in  the  canals  there ;  all  made  in  a  similar  manner  to  that  described 
in  article  201,  and  represented  in  figures  1  and  2,  plate  XV.  Their  principal  dimen- 
sions and  the  quantities  of  water  usually  gauged  in  them  are  as  follows :  — 

The  Merrimack  flume,  about  100  feet  long  and  50  feet  wide,  intended  to  gauge 
about  1,500  cubic  feet  of  water  per  second. 

The  Appleton  flume,  about  150  feet  long  and  50  feet  wide,  intended  to  gauge 
about  1,800  cubic  feet  of  water  per  second. 

The  Lowell  Manufacturing  Company's  flume,  about  150  feet  long  and  30  feet 
wide,  intended  to  gauge  about  500  cubic  feet  of  water  per  second. 

The  Middlesex  flume,  about  150  feet  long  and  20  feet  wide,  intended  to  gauge 
about  260  cubic  feet  of  water  per  second. 

The  Prescott  flume,  about  180  feet  long  and  66  feet  wide,  intended  to  gauge 
about  2,000  cubic  feet  of  water  per  second. 

The  Boott  flume,  about  100  feet  long  and  42  feet  wide,  intended  to  gauge  about 
800  cubic  feet  of  water  per  second. 

The  depths  of  the  water  in  these  flumes  are  various,  usually,  however,  between 
eight  and  ten  feet;  sometimes,  when  the  river  is  low,  the  depth  is  diminished  to 
about  six  feet. 

It  will  be  seen  that  the  widths  of  the  flumes  are  not  strictly  in  proportion 
to  the  quantities  of  water  intended  to  be  gauged  in  them ;  the  widths  and  depths 
have  usually  been  determined  by  the  dimensions  of  the  canals  in  which  they  are 
placed. 

240.  Under  the  existing  arrangements  at  Lowell,  a  daily  account  is  usually 
kept  of  the  excess  of  water,  if  any,  drawn  by  each  manufacturing  company,  over 
and  above  the  quantity  to  which  it  is  entitled  under  its  lease.  In  ordinary  times 
this  is  arrived  at  with  sufficient  exactness  by  means  of  occasional  measurements, 
but  when  the  flow  of  water  in  the  river  is  too  small  to  supply  the  wants  of 
all,  it  is  necessary  to  make  frequent  measurements  of  the  -quantity  of  water  drawn 
by  those  who  habitually  draw  an  excess.  In  the  latter  case  the  usual  course  of 
proceeding  is  this.  A  gauging  party,  consisting  of  one  or  more  engineers  and  a 
sufficient  number  of  assistants,  is  assigned  to  each  flume  where  measurements  are 
required.  Arrangements  are  made  sot  that  the  observations  for  a  single  gauge 
occupy  about  half  an  hour.  Several  gauges  are  made  during  the  day,  the  intervals 
between  the  times  when  the  observations  are  made  being  occupied  by  the  same 
party  in  working  out  the  results,  which,  as  soon  as  obtained,  are  communicated 
to  the  proper  local  authorities  at  the  manufacturing  establishments  where  the  water 
is  drawn.  This  is  done  to  enable  them  to  adjust  the  amount  of  machinery  they 
run,  so  as  to  draw  only  the  quantity  of  water  to  which  they  are  entitled.  If 


IN   OPEN   CANALS   OF   UNIFORM    RECTANGULAR   SECTION.  203 

they  continue  to  draw  an  excess  after  due  notice,  they  ure  liable  to  heavy  penal- 
ties. It  is  essential  to  the  proper  working  of  these  arrangements  that  the  results 
of  the  gaugings  should  be  arrived  at  and  communicated  as  speedily  as  possible ; 
with  this  view,  as  well  as  to  reduce  the  expense,  engraved  diagrams  and  printed 
forms  and  tables  have  been  prepared,  and  all  the  apparatus  provided  and  prep- 
arations made  which  can  in  any  way  facilitate  the  operation. 

241.  The  'mode  of  making  the  observations  for  a  gauge  in  a  measuring   flume 
is    substantially    the    same    as    that    practised    in     the    experimental    flume    in    the 
Tremont  Canal,  and   fully   described   in   articles   204  and  205.     With  the  view,  how- 
ever,  of   reducing    the    number   of  assistants   required,   a   stop-watch   beating   quarter 
seconds   is    used    instead    of    a    marine    chronometer,   and    the    electric   telegraph   is 
dispensed   with.      The  observer  with  the   stop-watch  takes  his  position  at  the   upper 
transit  station,  and  starts  the  watch  when  the  tube  passes   it ;   he  then  walks  to  the 
lower   transit  station  and  stops  the  watch  when  the   tube  passes  it.     By  this  method 
two  observers  are  dispensed  with.     Another  observer  notes  the  depth  of  the  water  in 
the    flume,    and   also    records   the    distances   of    the    tubes    from   the  left   side   of  the 
flume,  which   are    observed  and   called   by   the    assistants   who   put  in   and   take   out 
the   tubes.     One   other   assistant  is   required   to    carry   back    the    tubes   to    the    up- 
stream   station,   making   five    in    the    party. 

242.  Ordinarily,   about   an   hour   is   occupied  in    making   the    observations   for  a 
measurement.     The   following   measurement   is   given  in  detail  as  an  example  of  the 
•whole    process.      The    flume    in    which    it   was    made    is    situated   a   short   distance 
below   a   bend   of  about  ninety  degrees   in  the  canal,  which   produces   a   great  irreg- 
ularity  in    the    current,   the   velocity   being    much    greater   on    the    right-hand    side 
of    the    flume   than    on   the   left-hand   side ;    sometimes    there   is   no   sensible    current 
on   the   left-hand    side.      It   being    inconvenient   to   perform  the  measurement    under 
such   circumstances,  the   difficulty  was    remedied   by   placing  an   obstruction  near   the 
lower   end    of   the    bend;    the    up-stream    face    of    this   obstruction   was   an   oblique 
plane,   so    placed    as   to  direct   a   part  of  the  current   towards    the   leftrhand   side    of 
the    flume.      Although   far   from   producing   a   uniform   velocity   in   all   parts    of    the 
flume,  it  removed    all   the    trouble   in   making  the    measurement  due    to  the  original 
irregularity.      The    remaining   irregularities   in   the   velocity  are    indicated  by    the   in- 
flections  of  the   curved   line   A  B  on   plate   XIX. 

The    mean    width    of   the    part    of   the    flume    between    the    upper  and   lower 
transits   is   41.76   feet. 


204 


A   METHOD   OF   GAUGING  THE   FLOW   OF   WATER. 


TABLE    XXVI. 

GAUGE   OF   THE   QUANTITY   OF   WATER   PASSING   THE  BOOTT  MEASURING   FLUME,   MADE 
BETWEEN  10  HOURS  30  MINUTES  AND  11  HOURS  30  MINUTES,  A.M.,    MAY  I?TH,  1860. 


Time  during 

Distance  of  the  tube  from  the 

Distance  from 

which  the  tube 

left-hand  side  of  the  flume  during 

thi>  left-hand 

Length 

passed  from 

its  passage. 

Bide  of  the 

of  the 

the  up-stream 

Mean 

flume  at 

immeraec 

transit  station 

Telocity  of 

Depth  of 

which  the 

part  of  the 

to  the 

the  tube. 

At  the 

At  the 

water  in  the 

tube  was  put 

tube. 

down-stream 

upper 

lower 

flume. 

into  the 

transit  station 

transit 

transit 

Mean. 

water. 

a  distance  of 

station. 

station. 

70  feet. 

Feet. 

Feet. 

Seconds. 

Feet  per 

Feet. 

Feet. 

Feet. 

Feet. 

Second. 

0.0 

8.40 

33.3 

2.102 

0.3 

0.8 

0.55 

8.510 

1.5 

tt 

31.0 

2.258 

1.8 

1.6 

1.70 

8.481 

3.0 

ft 

30.2 

2.318 

3.2 

2.1 

2.65 

8.450 

4.5 

tt 

28.3 

2.473 

4.4 

4.5 

4.45 

8.470 

6.0 

u 

29.5 

2.373 

6.2 

5.4 

5.80 

8.445 

7.5 

u 

27.0 

2.593 

8.2 

10.1 

9.15 

8.438 

9.0 

tt 

26.2 

2.672 

9.7 

10.4 

10.05 

8.440 

10.5 

U 

25.0 

2.800 

10.5 

8.8 

9.65 

8.470 

12.0 

11 

25.8 

2.713 

12.3 

10.9 

11.60 

8.483 

13.5 

u 

25.2 

2.778 

13.8 

15.5 

14.65 

8.490 

15.0 

u 

25.0 

2.800 

15.2 

18.0 

16.60 

8.500 

16.5 

it 

29.5 

2.373 

17.0 

20.4 

18.70 

8.498 

18.0 

u 

27.0 

2.593 

18.0 

17.8 

17.90 

8.505 

19.5 

u 

28.8 

2.431 

19.7 

19.0 

19.35 

8.505 

21.0 

tt 

30.7 

2.280 

21.1 

20.9 

21.00 

8.522 

22.5 

tt 

81.8 

2.201 

23.4 

29.3 

26.35 

8.533 

24.0 

tl 

33.7 

2.077 

23.7 

22.1 

22.90 

8.510 

25.5 

If 

33.8 

2.071 

26.5 

29.7 

28.10 

8.495 

27.0 

tt 

31.0 

2.258 

27.0 

25.2 

26.10 

8.483 

28.5 

tt 

31.0 

2.258 

28.6 

26.5 

27.55 

8.495 

30.0 

tt 

29.0 

2.414 

31.0 

34.3 

32.65 

8.550 

31.5 

tt 

28.0 

2.500 

32.1 

30.0 

31.05 

8.630 

33.0 

u 

31.0 

2.258 

32.5 

28.1 

30.30 

8.610 

34.5 

u 

26.2 

2.672 

34.6 

36.7 

35.65 

8.625 

36.0 

u 

28.8 

2.431    . 

36.5 

35.0 

35.75 

8.632 

37.5 

tt 

28.5 

2.456 

37.5 

35.5 

36.50 

8.612 

39.0 

tt 

28.0 

2.500 

40.1 

40.5 

40.30 

8.578 

40.0 

tt 

28.0 

2.500 

39.0 

39.6 

39.30 

8.578 

41.0 

tt 

29.2 

2.397 

41.2 

40.6 

40.90 

8.560 

0.0 

tt 

34.3 

2.047 

0.5 

0.4 

0.45 

8.471 

10.0 

tt 

26.5 

2.642 

9.8 

8.7 

9.25 

8.580 

20.0 

tf 

32.2 

2.174 

20.9 

19.9 

20.40 

8.605 

30.0 

tt 

30.8 

2.273 

31.5 

33.8 

32.65 

8.635 

41.0 

tt 

30.5 

2.295 

41.4 

40.6 

41.00 

8.610 

Mean 

8.5294 

Products  of  the  mean  Telocities  into  the 
widths,  for  each  foot  in  width,  excepting  the 
last  product,  which  is  for  a  width  of  0.76 
feet ;  commencing  at  the  left-hand  side  of 
the  flume. 


2.264  X  0.76 


2.073 
2.193 
2.284 
2.359 
2.422 
2.478 
2.529 
2.577 
2.623 
2.666 
2.705 
2.744 
2.776 
2.801 
2.8H 
2.798 
2.747 
2.648 
2.514 
2.363 
2.249 
2.172 
2.120 
2.098 
2.105 
2.130 
2.163 
2.023 
2.246 
2.289 
2.331 
2.373 
2.418 
2.450 
2.483 
2.510 
2.531 
2.544 
2.540 
2.504 
2.417 
=  1.721 


Sum, 


101.523 


Mean  Telocity  )  101-623  _ 
of  the  tubes,  I  "  ~ 


IN  OPEN   CANALS   OF   UNIFORM  RECTANGULAR   SECTION.  205 

243.  The  mean  velocity  of  the  tubes  is  obtained  by  means  of  a  diagram,  a 
copy  of  which,  on  the  same  scale  as  the  original,  is  given  in  plate  XIX.  The 
small  circles  represent  the  several  observations,  the  abscissa  and  ordinate  of  each 
being  the  mean  distance  from  the  left-hand  side  of  the  flume  and  the  observed 
velocity  of  the  tube  as  given  in  table  XXVI.  The  curved  line  represents  the 
mean  and  is  drawn  by  the  eye,  giving  due  weight  to  each  observation.  The 
mean  velocity  is  2.4311  feet  per  second,  and  is  found  by  taking  a  mean  of  the 
ordinates  of  the  curve  ;  the  process  is  given  in  column  A,  table  XXVI. 

The  mean  depth  of  the  water  in  the  flume  was      .....     8.5294  feet. 
The  length  of  the  immersed  part  of  the  tube  was      ....     8.4000     " 
Difference     ...........  -  ......     .     0.1294     « 


Then  D  (art.  232)  =  =  0.0152. 

The  mean  section  of  the  water-way  in  the  flume  was 

41.76  X  8.5294  =  356.188  square  feet 

And  the  quantity  of  water  passing,  by  the  tube  measurement,  was 
356.188  X   2.4311  —  865.929  cubic  feet  per  second  =  #". 

This  is  to  be  corrected  by  formula  (2.),  art.  233. 

Substituting  for  D  its  value  0.0152,  -we  have  for  the  coefficient  of  correction 
1  _  0.116  (v/^>  —  0.1)  =  0.99730  (see  table  XXIX.)  and  the  corrected  quantity 
Q'"  =  0.99730  X  865.929  =  863.59  cubic  feet  per  second. 

244.  In  the  preceding  example  the  entire  volume  of  water  flowing  through 
the  canal  was  gauged.  It  often  happens  that  only  a  portion  of  the  entire  flow 
of  the  stream  is  to  be  gauged,  namely,  the  quantity  drawn  out  of  the  canal  at 
a  single  orifice  or  branch  canal.  In  this  case  a  flume  of  suitable  dimensions  is 
constructed  and  connected  with,  the  edges  of  the  orifice  or  the  sides  and  bottom 
of  the  branch  canal,  so  that  no  water  can  enter  the  orifice  or  branch  canal 
except  through  the  measuring  flume.  A  rough  preliminary  estimate  of  the 
quantity  should  be  made  by  some  other  method  ;  this  will  enable  the  sectional 
area  of  the  measuring  flume  to  be  determined,  so  that  the  velocity  in  it  may 
be  convenient  for  observation,  say  between  one  foot  and  three  feet  per  second, 
although  it  may  exceed  these  limits,  in  either  direction,  if  the  circumstances  are 
such  as  to  require  it.  It  will  generally  be  most  convenient  to  place  the  flume 
so  that  its  axis  will  be  parallel,  or  nearly  so,  with  the  axis  of  the  canal.  Its 


206  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER 

length  will  usually  be  limited  by  local  circumstances  and  economical  considerations; 
a  considerable  length  in  which  to  measure  the  velocity  of  the  loaded  tubes  is 
desirable,  although  not  essential.  If  the  means  for  observing  the  transits  and  the 
times  of  the  same  are  good,  a  less  length  is  necessary  than  in  cases  where  the 
means  of  observing  are  less  perfect.  By  means  of  the  electric  telegraph  and  a 
skilled  observer  of  the  chronometer,  as  in  the  experiments  at  the  Tremont  measur- 
ing flume  (art.  205),  an  interval  of  a  few  seconds  between  the  times  of  the 
transits  at  the  upper  and  lower  stations  will  enable  a  good  gauge  to  be  made. 
If  the  observations  are  made  in  the  less  perfect  manner  practised  at  the  Boott 
measuring  flume,  and  described  in  art.  241,  a  considerably  longer  interval  is 
necessary  in  order  to  attain  equally  accurate  results.  There  seems  to  be  scarcely 
any  limit  to  the  shortness  of  the  time  admissible  in  the  first  case,  if  corresponding 
care  and  precautions  are  adopted  in  making  the  observations.*  In  the  second  case, 
it  will  depend  much  on  the  degree  of  skill  of  the  observer.  The  method  has 
not  been  used  extensively  enough,  as  yet,  to  enable  a  limit  to  be  definitely  fixed. 
A  practised  observer,  with  a  stop-watch  beating  quarter  seconds,  the  transit  stations 
being  twenty-five  feet  apart,  has  been  able  to  observe  both  transits,  when  the  time 
between  them  was  ten  seconds,  and  in  some  cases  seven  and  a  half  seconds. 

245.  The    distance    between   the   transit   stations   is    only   a   part   of  the   length 
required   for   the  flume ;   a   certain   length  above   the    upper  transit  station   is   neces- 
sary  to    give   room   for   putting    the   tubes   into   the   water,   and   to   permit  them  to 
attain,   sensibly,   the   same    velocity   as    the   water   before    they   arrive   at   the    transit 
station.      By   reference    to    art.    195    it   will    be    seen    that    a    tube    two   inches   in 
diameter,   floating   twenty   feet,   attains    ff    of  the   velocity   of  the    current.     Twenty 
feet  was   about   the    distance   the    tubes   floated  before  they  reached  the  upper  tran- 
sit station,   in    the    experiments    given   in   table  XXII.,   from   which   the   formula   for 
the   correction   of  flume   measurements   was   determined,   and   the    correction   for   the 
very   small    error,    resulting    from    this    distance    being   insufficient,   is  implicitly   in- 
cluded in  the  formula.     Twenty  feet  may  therefore  be  taken  as  the  proper  distance, 
and   if   circumstances    are    such   as    to    require    a   much   less   distance,   the  resulting 
error  can  be  corrected  by  means  of  formula  (5.),  article  194. 

246.  The   same    method    may   be    extended    to    gauging    natural    watercourses. 
A  favorable    place    for    the    purpose    should    be    selected;    that    is,   one    free   from 
reverse    currents,   the   bottom   smooth,   the   section   uniform   for   a   sufficient   distance, 


*  Methods  for  making  and  recording  observations  of  time  are  practised  in  some  astronomical  obser- 
vatories, by  means  of  which  the  one-hundredth  part  of  a  second  is  estimated;  these  methods  could  un- 
doubtedly be  adapted  to  our  purpose  if  required. 


IN  OPEN   CANALS   OF   UNIFORM   RECTANGULAR   SECTION.  207 

and  with  as  long  a  reach  above,  free  from  bends,  great  irregularities  of  section 
and  obstructions,  as  can  be  found.  Two  parallel  sections,  in  planes  at  right  angles 
to  the  direction  of  the  current,  or  nearly  so,  should  be  carefully  measured,  so 
that  the  depth  at  every  point  may  be  known.  The  proper  distance  between  the 
sections  will  depend  much  on  the  regularity  of  the  channel ;  it  will  usually  be 
desirable  that  they  should  be  far  enough  apart  to  permit  the  observations  for  the 
velocity  to  be  made,  without  resorting  to  the  use  of  the  electric  telegraph;  except- 
ing in  very  large  rivers,  a  distance  of  from  fifty  to  one  hundred  feet,  depending 
on  the  width,  would  usually  permit  this  to  be  done  with  sufficient  accuracy  for 
most  purposes,  although  a  greater  distance  would  usually  be  desirable. 

The  loaded  poles  or  tubes  must  not  touch  the  bottom  while  passing  from  one 
transit  station  to  the  other.  It  will  probably  rarely  occur  that  one  hundred  feet 
in  length  of  the  channel  of  a  river  will  be  found  of  such  regularity  that  the  poles 
could  be  immersed  to  an  average  depth  of  six  inches  from  the  bottom.  By  resorting 
to  the  more  exact  mode  of  observing  the  transits,  the  sections  might  be  within 
twenty  feet  of  each  other,  or  even  half  that  distance  if  necessary.  There  would 
seldom  be  any  difficulty  in  finding  a  suitable  place  for  a  gauge  made  in  this 
manner,  in  any  river  confined  within  regular  banks.  Something  could  be  done,  in  so 
short  a  length,  towards  removing  obstructions  and  filling  up  depressions.  In  making 
the  observations,  loaded  tubes  or  poles,  of  lengths  adapted  to  the  different  parts  of 
the  section,  should  be  provided  ;  they  may  be  put  into  and  taken  out  of  the  water 
from  boats  or  rafts.  Theodolites  should  be  placed  in  the  planes  of  the  sections,  on 
the  same  bank ;  the  observer  at  each  should  have  the  key  of  a  break-circuit  within 
his  reach,  while  observing  the  transit  of  the  floating  pole.  The  observations  of  the 
times  of  the  transits  may  be  made  in  the  same  manner  as  at  the  Tremont  measur- 
ing flume  (art.  205).  If  the  sections  are  very  near  together,  a  separate  "observer 
may  be  necessary  for  the  transit  at  each  station,  both,  however,  using  the  same 
chronometer.  The  distance  from  fixed  points  on  the  bank,  at  which  the  floats  pass 
the  transits,  corresponding  to  the  distances  from  the  left-hand  side  of  the  flume,  in 
the  flume  measurements,  can  be  observed  by  means  of  marked  cords,  stretched 
across  the  river,  just  over  the  water,  and  at  short  distances  above  and  below  the 
sections,  and  supported  from  the  bottom  at  intervals,  if  necessary;  or  it  may  be 
done  by  means  of  a  system  of  signals  and  triangulations. 

The  section  of  the  river  not  being  rectangular,  it  will  usually  be  most  con- 
venient to  divide  it  into  several  parts,  finding  the  area  of  the  section,  the  mean 
velocity  of  the  poles,  computing  the  quantity  and  making  the  correction  by  formula 
(2.),  article  233,  for  each  part  separately.  The  sum  will  of  course  be  the  gauge  of 
the  whole  river. 


208  A  METHOD  OF  GAUGING  THE  FLOW  OF  WATER. 

The  degree  of  accuracy  attainable  in  gauging  a  natural  watercourse,  by  this 
method,  will  depend  entirely  upon  the  regularity  and  smoothness  of  the  part  of 
the  channel  selected  for  the  operation,  and  of  the  immediate  approach  to  the  same. 
If  the  bottom  is  covered  with  large  stones  or  sunken  timber,  it  will  prevent  the 
attainment  of  much  precision.  In  such  cases,  if  the  greatest  attainable  precision 
is  desired,  either  the  obstructions  must  be  removed  or  the  bed  of  the  channel 
filled  up  with  some  sort  of  material  suitable  for  the  purpose,  to  the  level  of  the 
top  of  the  obstructions.  In  any  case,  the  degree  of  precision  attainable  will  depend 
on  the  degree  of  approximation  in  the  channel  to  the  regularity  and  smoothness 
of  the  measuring  flumes. 


EXPERIMENTS  ON  THE  FLOW  OF  WATER  THROUGH  SUBMERGED  ORIFICES  AND 

DIVERGING  TUBES. 


247.  DANIEL   BERNOULLI   proved,  —  on    the    hypothesis    that    no  force  is   lost, — 
that  the  fluid  in  all  parts  of  the  same  section  has   the  same  velocity,  and  remains  in 
one    mass;    that    the   velocity   of   the    discharge    from    a    vessel,  by  an    orifice    of 
small   area   relatively   to    that    of   the   vessel,  is    that   due    to   the   head   above   the 
orifice   from   which   the   fluid  is   finally   discharged,   whether    such    orifice    is   in   the 
side    or    bottom    of    the    vessel    itself,   or   at  the    end    of   a    tube    projecting    from 
the   side   or  bottom   of  the   vessel,  the   sides   of  the  tube  being  either  parallel,  con- 
verging, or  diverging.*     This  being   established,   it  follows,   if  the   conditions   of  the 
hypothesis    can    be    complied    with,   that    the   velocity   of   discharge    from    a   simple 
orifice   in  a  vessel  may  be   increased   to   any   extent  by  the   application   of  a   tube 
with    diverging    sides;    for    the   area   of   the   orifice   at  the   end   of   the   tube    from 
which  the  fluid  is  finally  discharged  may  be  as  many  times  larger  than   the   orifice 
in   the   side   or    bottom    of   the   vessel    as    we    please,   and    as    the    same    quantity 
must  pass   through  both   orifices   in   the   same  time,  the  velocity  through  the  orifice 
in    the    vessel  will    be    as    much   greater  than   the   velocity  through   the   orifice   at 
the  end  of  the  tube  as  its  area  is  less. 

248.  The    fact    that    the    flow   through    an   orifice   could  be   increased   by   the 
application    of    a    diverging    tube    appears    to    have    been    known    to    the    ancient 
Romans.     Experiments  have    been  made  upon  them  in  modern  times  by  Gravesande, 
Bernoulli,    Venturi,    and    Eytelwein,   and    perhaps   others.      And    experiments   on   the 
discharge    of    air    between   two    discs,   which   afford   an    aperture   similar   in   effect   to 
a   diverging   tube,    have    been    made    by    Thomas    Hopkins.!     Most   of   our    exper- 
imental  knowledge  on   the  flow  of  water  through  diverging  tubes  is  due  to  Venturi, 
whose   experiments  were   made   at  Modena   about   the   year   1791,  and  published   in 

*  Hydrodynamica.     Strasburg,  1738. 

t  Memoirs   of  the   Literary   and  Philosophical  Society  of  Manchester.     Vol.  V.,   Second  Series.    Lon 
don,  1831. 

27 


210  EXPERIMENTS   ON  THE   FLOW   OF   WATER 

Paris  in  1797,  under  the  title,  Recherches  experimentales  sur  le  Principe  de  la  Com- 
munication laterale  du  Mouvement  dans  Us  Fluids* 

Venturi  experimented  on  many  forms  of  diverging  tubes;  in  pipes  of  regular 
form  the  maximum  increase  of  velocity  was  obtained  with  a  conical  tube  in  which 
the  sides  diverged  from  each  other  at  an  angle  of  4°  27';  this  tube  was  applied 
to  a  mouth-piece  having  nearly  the  form  of  the  contracted  vein;  a  certain  volume 
of  water  under  a  constant  head  was  discharged  through  the  mouth-piece  alone 
in  forty-two  seconds;  when  the  diverging  tube  was  applied  to  the  mouth-piece, 
the  same  quantity  of  water  was  discharged,  under  the  same  head,  in  twenty-one 
seconds;  increasing  the  velocity  through  the  mouth-piece  in  the  ratio  of  1  to  2. 
In  a  similar  tube  of  greater  length  the  water  did  not  fill  the  tube  throughout 
its  whole  length  unless  a  prominence  was  made  on  the  inside  of  the  tube,  at  the 
bottom,  which  caused  the  water  to  fly  upward  and  fill  the  down-stream  end  of 
the  tube;  with  this  tube,  the  same  volume  of  water  was  discharged  in  nineteen 
seconds,  increasing  the  discharge  through  the  mouth-piece  in  the  ratio  of  1  to 
2.21. 

Eytelwein  made  some  similar  experiments  with  a  mouth-piece  and  a  tube 
whose  sides  diverged  at  an  angle  of  5°  9'.  He  found  that  the  application  of  the 
tube  to  the  mouth-piece  increased  the  velocity  through  the  latter  in  the  ratio 
of  1  to  1.69. 

249.  According  to  Bernoulli's  theory,  in  Venturi's  experiment,  last  above 
quoted,  the  velocity  through  the  smallest  section  of  the  mouth-piece  should  be  in- 
creased by  the  diverging  tube,  in  the  ratio  of  1  to  3.03.  In  Eytelwein's  exper- 
iment the  increase  should  be  in  the  ratio  of  1  to  3.21.  In  both  these  experiments 
the  water  in  the  tube  undoubtedly  remained  in  unbroken  masses.  There  must,  con- 
sequently, have  been  considerable  losses  of  force.  The  increased  flow  appears  to  be 
due  to  what  is  termed  by  Venturi  the  lateral  communication  of  motion  in  fluids, 
and  to  the  pressure  of  the  atmosphere.  According  to  the  principle  of  Venturi,  a 
column  of  water  flowing  through  a  mass  of  water  at  rest  tends  to  communicate 
a  portion  of  its  velocity  to  the  mass,  and  to  cause  it  to  move  along  with  it;  and 
if  the  column  of  water  is  moving  in  a  pipe  a  little  larger  than  itself,  it  will 
communicate  motion  to  the  entire  shell  of  water  surrounding  it.  If  the  water  is 
flowing  through  a  conical  tube  whose  sides  diverge  at  a  small  angle,  the  section 
of  the  pipe  is  continually  enlarging  by  insensible  degrees;  but  by  the  principle 
of  Venturi  the  stream  must  fill  each  successive  section,  and  the  mean  velocity 

-I 

*  See    a    translation    of   Venturi's    work,    in    Nicholson's    Journal    of    Natural    Philosophy,   Vol.   III. 
London,  1802.     Also,  in  Tracts  on  Hydraulics,  by  Thomas  Tredgold,  2d  Edition.     London,  1836. 


THROUGH   SUBMERUED  ORIFICES  AND   DIVERGING  TUBES.  211 

must  diminish  in  the  ratios  that  the  areas  of  the  sections  increase.  The  pressure 
of  the  atmosphere  on  the  surface  of  the  water  in  the  vessel  and  on  the  orifice 
from  which  the  water  escapes  may  for  this  purpose  be  called  the  same,  and  equal 
to  a  column  of  water  thirty-four  feet  high.  Supposing  the  mass  of  water  flowing 
through  the  pipe  to  be  divided  into  very  thin  slices,  by  planes  at  right  angles 
to  the  direction  of  the  current ;  from  its  inertia,  each  slice  will  tend  to  retain  its 
velocity,  but  on  account  of  the  enlarging  sections  it  cannot  do  this,  but  tends  to 
separate  itself  from  the  slice  immediately  following  it;  this  is  prevented  by  the 
pressure  of  the  atmosphere,  and  the  effect  is  to  balance  a  portion  of  the  pressure  of 
the  atmosphere  on  its  down-stream  side ;  the  entire  pressure  of  the  atmosphere 
remains  on  the  up-stream  side  of  the  slice,  and  the  difference  between  the  effective 
pressures  on  the  up-stream  and  down-stream  sides  accelerates  the  motion  of  the 
slice.  All  the  slices  are  acted  on  in  a  similar  manner,  and  the  increased  discharge 
is  due  to  the  sum  of  the  actions  upon  them. 

In  the  experiment  above  quoted  of  Venturi,  with  a  pipe  of  regular  form,  the 
discharge  through  the  orifice  took  place  under  a  head  of  2.887  feet;  the  head  being 
as  the  square  of  the  velocity,  the  equivalent  head,  under  which  the  discharge  took 
place  with  the  diverging  tube,  was  2.887  X  22  =  11.548  feet,  which  exceeds  the 
actual  head  of  water  in  the  experiment  by  8.661  feet,  which  is  the  portion  of  the 
total  pressure  of  the  atmosphere  on  the  surface  of  the  water  in  the  reservoir  ren- 
dered active  in  that  experiment. 

250.  Venturi  found  no  increased  discharge  by  increasing  the  length  of  his 
diverging  tube  beyond  1.096  feet,  on  account  of  the  water  not  filling  the  whole 
section  of  the  part  of  the  tube  added  beyond  that  length.  This  difficulty,  however, 
can  be  obviated  by  submerging  the  diverging  tube ;  for  in  that  case  it  must 
remain  full  of  water,  whatever  may  be  its  length  or  the  angle  of  divergency  of  its 
sides. 

In  these  experiments  the  tubes  were  submerged,  which  distinguishes  them  from 
any  previously  recorded,  and  greater  effects  were  produced.  The  diffuser  applied 
by  Mr.  Boyden  to  turbine  water-wheels,  to  increase  their  efficiency  (art.  12),  acts 
on  the  same  principle  as  the  diverging  tube ;  this  apparatus  has  been  extensively 
applied  in  Lowell,  and  it  has  thus  become  a  matter  of  great  interest  to  ascertain 
to  what  extent  a  conical  diverging  tube,  discharging  under  water,  could  be  made 
to  increase  the  discharge  through  a  simple  orifice.  For  this  purpose  the  following 
experiments  were  made. 


212  EXPERIMENTS   ON  THE   FLOW   OF   WATER 


DESCRIPTION  OF  THE  APPARATUS. 

251.  The  tube  used  in  these  experiments  is  represented  by  figures  1,  2,  3,  and 
4,  plate  XXI.     It   is  composed   of  cast  iron  and  is  made  in  five  pieces,  A,  S,  C,  D, 
and  E,  which  when  screwed  together,  as  represented  in  figures  1  and  2,  form  a  com- 
pound tube,  consisting  of  a  mouth-piece  of  a  form  to  avoid  contraction,  and  a  diverg- 
ing tube,  in  which  the  diameter  increases  from  about  0.1  foot,  at  its  junction  6  with 
the  mouth-piece,  to  about  0.4  foot  at  /.     The  part  of  the  mouth-piece  between  a  and 
g  is  formed  by  the  revolution  of  a  common  cycloid  about  the  axis  of  the  tube ;   from 
a  to  &  it  is  cylindrical.     The  interior  of  the  parts  C,  D,  E  are  frustums  of  a  cone; 
a  portion  of  the  part  B  is  also  a  frustum  of  the  same  cone ;  but,  to  avoid  any  angle 
in   passing  from   the   cylinder  a  6  to  the  frustum  of  the  cone,  a   portion  of  the  part 
B  is  formed   by   the   revolution   of  an  arc   of    a   circle   of   about  22.69   feet  radius, 
the  sides  of  the  cylinder  a  6  and  of  the  cone  both  being  tangent  to  this  arc.     The 
parts   of  the   compound   tube   being   screwed   together   could  be  readily  taken   apart 
and    the    mouth-piece   used  by    itself,    or   with   one    or  more    of  the    conical    parts 
attached.      The   interior   of  the    mouth-piece    and    diverging    tubes   were   first    turned 
separately,  they  were  then  screwed  together   and   ground   on  a  mandril  with   emery 
until  they  became  quite  smooth,  without,  however,  having  a  bright  polish.     This  mode 
of  finishing   insured   the   smallest  possible   degree   of  irregularity  at  the  junctions  of 
the  several  parts. 

252.  For    the    purpose    of   making    the    experiments,   the   compound    tube   was 
mounted   in   a   cistern   (figures  1,  2,  and  3,    plate   XX.)   constructed  for  the    purpose. 
The   cistern  was  made   of  white-pine  wood,  very  strongly  framed,  and  supported  on 
brick   piers,  which   were    built  up    several    feet    in    height    from    a    solid    foundation 
The    cistern    consists    of    three    compartments;    the    upper    compartment,    E,   is    the 
reservoir  supplying  the  mouth-piece  M,  and   the   diverging   tube   attached   to   it.     F, 
the  middle  compartment,  receives  the  water  discharged  through  the  tube.      G  is   the 
lower  compartment,  in  the  end  of  which  is  placed  the  weir,  W,  at  which  the  quantity 
of  water  discharged  was  gauged. 

The  supply  of  water  for  the  experiments  was  obtained  from  the  main  pipes 
laid  down  by  the  manufacturing  companies  at  Lowell  for  conveying  water  from  an 
elevated  reservoir  to  their  several  establishments  mainly  for  the  purpose  of  extin- 
guishing fire.  For  these  experiments,  it  was  important  that  the  supply  of  water 
flowing  into  the  reservoir  E  should  be  as  nearly  uniform  as  possible,  but  the  effec- 
tive pressure  in  the  main  pipes  was  subject  to  some  irregularity,  which  of  course 


THROUGH   SUBMERGED   ORIFICES   AND   DIVERGING  TUBES.  213 

caused  a  corresponding  irregularity  in  the  discharge  from  the  orifice  through 
which  the  supply  of  water  was  drawn.  To  eliminate  this  source  of  irregularity, 
the  water  was  first  drawn  into  the  cistern  I,  figures  2  and  3,  plate  XX.,  in  con- 
siderably greater  volume  than  was  required  to  be  admitted  into  the  reservoir  E ; 
the  excess  passed  over  a  weir  in  the  side  of  the  cistern  /,  and  from  thence  was 
discharged  through  the  waste-pipe  K.  The  supply  for  the  reservoir  E  was  drawn 
from  the  cistern  /  through  the  pipe  H,  the  quantity  being  regulated  by  the  cock 
L.  By  this  arrangement,  it  will  be  seen  that,  so  long  as  the  water  was  admitted 
into  the  cistern  I  in  excess  of  that  admitted  into  the  reservoir  E,  the  head  acting 
on  the  cock  L  must  have  been  subject  to  only  very  small  variations,  and  con- 
sequently the  discharge  through  a  constant  orifice  in  the  cock  L  must  have  been 
very  nearly  uniform.  It  was  important  that  the  water  in  the  part  of  the  reservoir 
E,  near  the  side  containing  the  mouth-piece,  should  be  as  nearly  quiescent  as  pos- 
sible. The  water  was  admitted  under  a  head  of  about  18  feet,  which  necessarily 
produced  a  great  commotion  in  the  part  of  the  reservoir  where  it  entered,  and  to 
prevent  this  from  extending  to  the  side  containing  the  mouth-piece,  it  was  made  to 
pass  through  six  diaphragms,  R,  R,  R,  &c.,  figures  1  and  2,  plate  XX.  The  first  two 
diaphragms  were  made  of  boards,  about  one  inch  thick,  containing  numerous  holes 
about  half  an  inch  in  diameter,  as  shown  in  figure  4;  the  other  four  diaphragms 
were  of  strainer-cloth,  placed  about  two  inches  apart  and  stretched  tightly  in  a  frame. 
The  strainer-cloth  used  was  the  well-known  fabric  sold  under  that  name,  made  of 
flax  or  hemp,  with  about  twenty  threads  to  an  inch  in  both  warp  and  filling,  the 
width  of  the  spaces  between  the  threads  being  from  two  to  three  times  the  thickness 
of  the  thread.  The  effect  of  these  diaphragms  was  to  prevent  any  sensible  commotion 
in  the  part  of  the  reservoir  between  the  lower  diaphragm  and  the  side  containing  the 
mouth-piece.  The  part  of  the  reservoir  E,  between  the  down-stream  diaphragm  and 
the  mouth-piece,  was  about  2.34  feet  long  in  the  direction  of  the  current,  3  feet 
wide,  and  4.5  feet  deep.  The  division  F  was  about  6.75  feet  long,  3  feet  wide, 
and  3.35  feet  deep;  the  water  passed  from  this  division  to  the  division  G  through 
the  diaphragm  N,  similar  to  the  wooden  diaphragms  in  division  E,  above  described; 
and  also  through  the  diaphragm  P,  consisting  of  a  single  thickness  of  strainer- 
cloth.  The  dimensions  of  the  part  of  the  reservoir  G,  between  the  diaphragm  P 
and  the  end  containing  the  weir  W,  is  about  3.6  feet  long  in  the  direction  of  the 
current,  3  feet  wide,  and  3.20  feet  deep.  The  disturbance  in  the  division  F  was 
slight,  and  as  the  apparatus  was  first  designed,  the  weir  was  placed  in  the  partition 
N~,  but  on  trial  the  agitation  was  found  to  be  too  great  to  admit  of  an  un- 
exceptionable gauge  at  the  weir;  the  division  G  was  then  added,  which,  with 
the  diaphragms,  removed  all  difficulty  from  this  cause. 


214  EXPERIMENTS  ON  THE   FLOW  OF   WATER 

253.  A  weir  was  adopted  to  gauge  the  quantity  of  water  passing  through 
the  tube,  in  preference  to  any  other  kind  of  orifice,  because  it  admitted  of 
greater  variations  in  the  quantity  of  water  discharged,  with  any  admissible  variation 
in  the  height  of  the  water  in  the  reservoir  in  the  side  of  which  it  is  placed ; 
and  by  adopting  a  weir  of  the  same  dimensions  and  form  as  that  used  by 
Poncelet  and  Lesbros  (art.  161),  the  quantity  could  be  computed  with  great 
precision. 

The  weir  W,  figures  1,  2,  and  6,  plate  XX.,  is  represented  on  a  larger  scale 
by  figures  5,  6,  and  7,  plate  XXL,  and  a  section  of  the  crest  of  the  weir  is 
given,  full  size,  in  figure  8,  plate  XXI.  The  crest  and  sides  of  the  weir  were 
made  of  plates  of  cast  iron,  planed  and  finished  with  great  care,  the  up-stream 
edges  presented  to  the  current  having  sharp  corners,  or  as  nearly  so  as  could 
be  made  with  that  metal.  The  only  material  variation  from  the  weir  used  by 
Poncelet  and  Lesbros  is  in  the  thickness  of  the  crest,  which  in  their  weir  was 
an  edge,  whereas  in  our  weir  it  had  a  thickness  of  about  0.02  inch ;  this  variation 
was  made  to  enable  the  zero  points  of  the  several  gauges,  used  for  measuring  the 
heights  of  the  water  in  the  different  compartments  of  the  apparatus,  to  be  made 
in  a  particular  manner,  which  will  be  described  hereafter.  This  difference  in  the 
thickness  of  the  crest  of  the  weir  could  have  affected  the  accuracy  of  the  gauge 
in  only  a  few  of  the  experiments,  namely,  those  in  which  the  depth  on  the  weir 
was  less  than  0.05  foot,  as  at  this  depth  and  all  greater  depths  it  was  observed 
that  the  contraction  was  complete ;  that  is  to  say,  at  this  depth  the  stream 
passing  over  the  weir  touched  the  orifice  only  at  the  up-stream  edge,  as  repre- 
sented in  figure  3,  plate  XFIII.,  and  the  flow  was  the  same  as  if  the  crest  of 
the  weir  had  no  sensible  thickness.  With  depths  on  the  weir  less  than  0.05  foot, 
the  stream  of  water  was  in  contact  with  the  whole  width  of  the  crest  of  the 
weir ;  which,  if  it  had  any  sensible  effect,  would  tend  to  increase  the  discharge, 
with  the  same  depth  on  the  weir,  in  consequence  of  an  action  similar  to  that 
produced  by  a  short  additional  tube  attached  to  the  down-stream  side  of  an 
orifice  in  a  thin  plate. 

The  length  of  the  curved  part  of  the  mouth-piece  A,  figure  2,  plate  XXL, 

measured  on   the   axis  a  g,  is 1.00  foot 

The  length   of  the  cylindrical  part  of  the   mouth-piece,   measured  on   the 

axis  a  &,  is 0.10     " 

The  effective  lengths  of  the  parts  S,  C,  D,  and  E,  of  the  diverging  tube, 

are  each 1.00     " 

The  diameter  of  the   circle  generating  the  semi-cycloid  of  the  mouth-piece 

is  .   0.635  « 


THROUGH  SUBMERGED  ORIFICES  AND  DIVERGING  TUBES.  215 

The  diameters  of  the  several  parts  of  the  mouth-piece  and  diverging 
tube  are  given  in  column  15,  table  XXVII. 

The  angle  of  divergence  of  the  sides  of  the  conical  part  of  the  com- 
pound tube  is  , 5"  1'. 

254.  The  elevations  of  the  surface  of  the  water  in  the  several  compartments 
of  the  cistern  were  measured  by  means  of  the  hook  gauges  represented  by  figures 
9,  10,  and  11,  plate  XXL,  and  described  in  articles  45  and  143.  They  were 
placed  in  the  hook  gauge  boxes  A,  B,  C,  D,  figures  1  and  2,  plate  XX.  Com- 
munication was  established  between  the  several  hook  gauge  boxes  and  the  cor- 
responding compartments  of  the  cistern  by  means  of  the  orifices  0,  figures  1,  2, 
5,  and  6.  The  orifices  affording  communication  with  the  compartments  F  and  G 
were  0.10  foot  in  diameter ;  the  orifice  affording  communication  with  the  compart- 
ment E  was  about  five  times  as  large;  oscillations  in  the  elevation  of  the  surface 
being  anticipated  in  this  compartment,  the  amplitude  of  which  it  was  desirable 
to  measure.  There  is  reason  to  think  that  the  flow  through  a  diverging  tube  is 
to  a  certain  extent  in  a  condition  of  "unstable  equilibrium.  In  Venturi's  exper- 
iments, the  water  discharging  into  the  air  from  diverging  tubes  was  observed  to 
have  great  irregularity  of  motion,  "  and  even  eddies  within  the  tube ;  whence  the 
jet  comes  forth  by  leaps,  and  with  irregular  scattering."  *  These  irregularities  are 
undoubtedly  due,  in  part  at  least,  to  an  unstable  equilibrium,  and  there  must  be 
a  corresponding  irregularity  in  the  exhausting  power  of  the  diverging  tube,  which 
would  be  indicated,  in  our  experiments,  by  oscillations  in  the  elevation  of  the 
surface  of  the  water  in  compartment  E,  which  would  rise  and  fall  as  the 
exhausting  power  of  the  tube  was  less  or  greater. 

The  elevations  of  the  surface  of  the  water  in  all  the  compartments  is 
reckoned  from  the  top  of  the  weir.  When  no  water  was  admitted  to  the  reser- 
voir E,  the  water  in  all  the  divisions  of  the  cistern  would  fall  to  the  level  of 
the  crest  of  the  weir.  The  comparison  between  the  zero  points  of  the  several 
hook  gauges  and  the  crest  of  the  weir  was  made  in  the  manner  described  in 
article  143.  Two  ten-pointed  instruments  (figure  14,  plate  XXI.),  of  slightly  dif- 
ferent dimensions,  were  used,  which  furnished  independent  results,  a  mean  of  which 
was  taken.  They  were  made  of  steel  and  magnetized,  which  enabled  them  to 
maintain  their  positions  when  placed  on  the  crest  of  the  weir.  Small  variations 
in  the  apparatus  were  expected  to  occur,  resulting  from  changes  of  temperature 
and  in  the  hygrometric  condition  of  the  wood  of  which  the  cistern  was  con- 

*  Tracts   on   Hydraulics. 


216 


EXPERIMENTS   ON   THE   FLOW   OF   WATER 


structed;    comparisons   were   accordingly   made   on   each   day   that  experiments  were 
made ;   the  results  are  given  in  the  following  table :  — 


Date. 

Corrections  to  be  applied  to  the  reading  of  the  hook  gauges,  to  give  the 

elevations  of  the  points  of  the  hooks  above  the  top  of  the  weir. 

1851. 

Gauge  .1. 

Gauge  B. 

Gauge  C. 

Gauge  D. 

September  20. 

—1.5535 

—1.5490 

—1.5451 

—0.3921 

"            21  A.M. 

—1.5519 

—  .5476 

—1.5439 

—0.3916 

"             21  P.M. 

—1.5525 

—  .5484 

—1.5449 

—0.3920 

22. 

—1.5528 

—  .5487 

—1.5447 

—0.3918 

"         25. 

—1.5531 

—  .5487 

—1.5454 

—0.3926 

«         26. 

—  1.5535 

—  .5490 

—1.5458 

—0.3930 

October        7. 

—1.5541 

—  .5502 

—1.5474 

—0.3940 

"           10. 

—1.5541 

—1.5502 

—1.5476 

—0.3938 

«           12. 

—1.5541 

—1.5502 

—1.5476 

—0.3942 

"           16. 

—1.5536 

—1.5500 

—1.5472 

—0.3935 

MODE   OF  CONDUCTING  THE  EXPERIMENTS. 

255.  Water  was    admitted    through   the   leathern  hose    Q    into    the    cistern  /, 
figures   2   and   3,   plate   XX.,  in   excess  of  the   supply  required  for  the  experiment. 
The   index   of  the   cock   L,   figures  2  and  3,  was  set  hi  the  desired   position.     When 
it   was    supposed   that   the   flow   had    become   permanent    throughout    all    the    divis- 
ions  of  the    cistern,   observations  of  the    elevations   of  the   surface   of   the  "water   in 
the  several  compartments  were  commenced;  they  were  taken  by  a  separate  observer 
at  each  hook  gauge,   every   thirty   seconds,   and  were   continued  until  some  minutes 
after    the    elevation    of   the    surface    in    the    compartments  F  and    G  had   become 
stationary,  which  indicated   that   a  permanent   flow  had  been  obtained.     The  watches 
used  by   the  several   observers    were    set   to   indicate  the   same   time,   and   the   time 
when   each  observation   was   made    being   recorded,   a   subsequent   comparison   of  the 
records   of  the   observations   made   at  the   several   hook  gauges  enabled   those  to  be 
selected    in    which    the    permanence    of    the   flow   was    the   most   perfect.      Not   less 
than   five,  and    usually   more    than   ten,   successive    observations,   made   at  the   same 
times    at    each    hook    gauge,  were    used,    from  which    the   mean  elevations  in   the 
several  compartments  during  the  experiment  were  deduced. 

256.  Experiments   1   to   18,   table   XXVII.,  were    made   with    the    mouth-piece 
A   alone.     Experiments  19  to  38  were   made  with   the   mouth-piece  A   and  the  first 
joint  B  of  the   diverging   tube.     Experiments  39  to  50  were  made  with  the  mouth- 
piece  and  the   two  joints   B  and    C  of  the   diverging  tube.     Experiments  51  to  64 


THROUGH    SUBMERGED   ORIFICES   AND   DIVERGING   TUBES.  217 

were  made  with  the  mouth-piece  and  the  three  joints  B,  C.  and  D  of  the  diverging 
tube.  Experiments  65  to  90  were  made  with  the  complete  compound  tube,  repre- 
sented by  figures  1  and  2,  plate  XXL,  and  in  figures  1  and  2,  plate  XX.  Exper- 
iment 91  was  made  with  the  mouth-piece  alone.  Experiment  92,  with  the  complete 
compound  tube.  Experiments  93  to  101  were  made  with  an  orifice  in  a  thin 
plate  represented  by  figures  12  and  13,  plate  XXI.  This  plate  is  of  cast  iron 
0.042  foot  in  thickness,  but  the  orifice  is  chamfered  off  on  the  down-stream  side, 
so  that  the  effective  thickness  of  the  plate  at  the  orifice  is  0.0014  foot,  or  about 
one  sixtieth  of  an  inch. 

257.  The  mouth-piece,  diverging  tubes,  and  plate  were  all  of  cast  iron ;  this 
metal  was  adopted  instead  of  brass  as  being  the  cheapest,  and  experience  having 
shown  that  oxidation  of  cast  iron  immersed  in  the  water  of  Merrimack  River  pro- 
ceeds very  slowly,  and  expecting  to  be  able  to  find,  readily,  some  substance,  a  coating 
of  which,  of  imperceptible  thickness  on  the  surface  of  the  metal,  would  entirely 
prevent  it;  no  such  substance  was  found,  however.  Drying  oils  of  several  kinds 
were  tried,  also  a  mixture  of  grease  and  mercury,  also  collodion,  but  without  satis- 
factory effect.  The  plan  finally  adopted  was  to  keep  the  interior  of  the  orifices  and 
tubes  and  the  accessible  parts  of  the  weir,  when  not  in  use,  covered  with  a  thick 
coating  of  grease.  Previous  to  each  session  of  the  experiments  this  was  removed  as 
completely  as  possible  by  rubbing  with  cotton-waste  and  woollen  cloth,  until  on  rub- 
bing with  a  clean  white  cloth  no  sensible  mark  was  made  on  it.  Of  course  the 
whole  of  the  grease  was  not  removed  by  this  operation ;  the  quantity  remaining,  how- 
ever, must  have  been  extremely  small,  but  it  was  sufficient  to  protect  the  iron  from 
oxidation  for  some  time,  or  until  it  was  partially  washed  off  With  this  process, 
however,  there  must  have  been  constant  changes  going  on  in  the  state  of  the 
interior  surface  of  the  tube,  which  might  affect  the  flow  of  the  water  in  some 
degree.  I  accordingly  noted  carefully  the  circumstances  and  indications  relating  to 
the  application  and  removal  of  the  grease;  and  under  the  head  of  Remarks  in  the 
table  of  experiments  I  have  stated  the  essential  parts  of  my  observations  on  this 
matter. 


28 


218 


TABLE 

EXPERIMENTS  ON  THE  FLOW   OF  WATER 


1 

2 

3 

4 

5 

G 

7 

8 

9 

Time  of  making  the  obser- 

Temperature, 

Reference  to 

Powtion 

Mean 

Value  of 

Quantity 

Height  of  the  surface  of  the 

vations  from   which  the 

in  degrees 

figure  2, 

of  the 

depth  of 

Cin  the 

of  water 

water  in  compartment  F, 

No. 

iiH'iin    heights   given 

of  Fahren- 

plate XXI., 

index  of 

water  on 

formula 

discharged, 

figures  1  and  2, 

Date. 

in  this  table  are 

heit's  ther- 

indicating 

the  in- 

the weir. 

in  the 

calculated 

plate  XX. 

of 

deduced. 

mometer  ; 

the  parts  of 

let  cock. 

by  gauge 

next  col- 

by  the 

the  com- 

A. 

uinii. 

formula 

the 

tsu 

pound  tube 
used. 

k 

Z>  = 

KXJI 

loot. 

Beginning. 

Ending. 

Clh\/  '2(ih 

by  gauge 
E. 

by  gauge 

Mean. 

II. 

Min. 

Sec. 

H. 

Min. 

Sec. 

of  the 
air. 

of  the 
water. 

Degrees. 

Feet. 

Cubic  feet 
per  second. 

Feet. 

Feet. 

Feet. 

1 

Sept.  20,  P.M. 

3 

37 

15 

3 

50 

45 

64.6 

A 

32.50 

0.0269 

0.4219 

0.00980 

0.0269 

0.0269 

0.0269 

2 

it               it              li 

3 

57 

0 

3 

59 

0 

64.6 

tf 

32.50 

0.0270 

0.4219 

0.00985 

0.0269 

0.0270 

0.0269 

3 

tt               it               U 

4 

22 

30 

4 

26 

45 

64.6 

tt 

34.25 

0.0388 

0.4202 

0.01690 

0.0391 

0.0392 

0.0391 

4 

U           it           it 

4 

31 

;)ii 

4 

35 

30 

64.6 

tf 

34.25 

0.0383 

0.4203 

0.01658 

0.0380 

0.0384 

0.0382 

5 

ti         it        tt 

4 

53 

15 

4 

58 

15 

64.6 

fi 

35.50 

0.0467 

0.4191 

0.02226 

0.0469 

0.0471 

0.0470 

6 

11            tt           U 

5 

20 

50 

5 

26 

0 

62.6 

64.6 

tt 

36.50 

0.0532 

0.4182 

0.02701 

0.0536 

0.0535 

0.0535 

7 

tt       tt       tt 

5 

38 

40 

5 

41 

50 

62.6 

64.6 

tf 

37.50 

0.0607 

0.4172 

0.03283 

0.0612 

0.0612 

0.0612 

8 

"       21,  A.M. 

9 

11 

40 

y 

17 

40 

56.4 

62.8 

ft 

37.50 

0.0616 

0.4170 

0.03355 

0.0614 

0.0620 

0.0617 

9 

it         it        it 

9 

41 

40 

9 

45 

40 

58.5 

62.9 

ft 

38.50 

0.0680 

0.4162 

0.03884 

0.0683 

0.0686 

0.0684 

10 

tt               it               it 

10 

H 

30 

10 

20 

0 

60.5 

63.2 

tt 

39.50 

0.0739 

0.4153 

0.04391 

0.0738 

0.0746 

0.0742 

11 

it               ti               ft 

10 

46 

50 

10 

54 

20 

60.9 

63.3 

If 

40.50 

0.0803 

0.4145 

0.04964 

0.0799 

0.0802 

0.0800 

12 

ti         it        ti 

11 

15 

40 

11 

19 

0 

61.0 

63.4 

tt 

41.50 

0.0848 

0.4138 

0.05378 

0.0846 

0.0852 

0.0849 

13 

"      21,  P.M. 

2 

16 

40 

2 

23 

50 

61.0 

63.4 

ft 

42.50 

0.0906 

0.4130 

0.05927 

0.0905 

0.0910 

0.0907 

14 

it                tt              ti 

2 

39 

30 

2 

45 

20 

62.0 

63.6 

tt 

43.25 

0.0945 

0.4125 

0.06306 

0.0948 

0.0951 

0.0949 

15 

tt         ti        tt 

3 

5 

0 

3 

9 

0 

62.9 

63.7 

tt 

44.00 

0.0991 

0.4118 

0.06761 

0.0992 

0.0999 

0.0995 

1C 

it         tt        tt 

3 

30 

30 

3 

33 

40 

67.7 

63.7 

U 

44.75 

0.1037 

0.4112 

0.07227 

0.1036 

0.1045 

0.1040 

17 

tt         it        tt 

3 

46 

40 

3 

52 

0 

67.1 

63.8 

tf 

45.50 

0.1069 

0.4108 

0.07556 

0.1071 

0.1077 

0.1074 

18 

"      22,  A.M. 

9 

13 

30 

9 

17 

20 

58.0 

62.7 

tf 

45.67 

0.1072 

0.4107 

0.07586 

0.1063 

0.1085 

0.1074 

19 

it        ti        tt 

10 

29 

15 

~10 

Ii4 

0 

60.9 

62.9 

AB 

54.50 

0.1505 

0.4049 

0.12441 

0.1531 

0.1559 

0.1545 

20 

"     25,  A.M. 

8 

59 

20 

9 

3 

30 

60.8 

62.4 

tt 

32.50 

00285 

0.4216 

0.01068 

0.0284 

0.0282 

0.0283 

21 

it        ti        ti 

!i 

16 

0 

9 

19 

0 

61.0 

62.3 

ft 

34.25 

0.0393 

0.4201 

0.01722 

0.0393 

0.0392 

0.0392 

22 

it        tt        tt 

9 

L'.S 

(i 

9 

31 

0 

61.3 

62.3 

ft 

35.50 

0.0472 

0.4190 

0.02261 

0.0478 

0.0476 

0.0477 

23 

tt        tt       tt 

9 

45 

30 

9 

45 

50 

61.7 

62.3 

ft 

36.50 

0.0555 

0.4179 

0.02876 

0.0556 

0.0557 

0.0556 

24 

ti         tt        it 

9 

58 

0 

10 

1 

25 

62.2 

62.3 

tf 

37.50 

0.0618 

0.4170 

0.03372 

0.0621 

0.0622 

0.0621 

25 

it         (i        tt 

10 

14 

10 

10 

17 

0 

63.8 

62.4 

ft 

38.50 

0.0678 

0.4162 

0.03867 

0.0682 

0.0686 

0.0684 

2G 

it         tt        ti 

10 

30 

0 

10 

33 

20 

63.7 

62.4 

(t 

39.50 

0.0732 

U.4154 

0.04330 

0.0740 

0.0740 

0.0740 

27 
28 

tt         tt        tt 

10 
11 

45 
5 

30 
30 

10 

11 

50 
8 

0 
20 

64.2 
64.8 

62.4 
62.5 

tt 

40.50 
41.50 

0.0796 
0.0849 

0.4146 
0.4138 

0.04900 
0.05387 

0.0803 
0.0860 

0.0805 
0.0865 

0.0804 
0.0862 

29 

tt        tt        tt 

11 

19 

10 

11 

25 

10 

65.0 

62.6 

tt 

42.50 

0.0901 

0.4131 

0.05880 

0.0911 

0.0914 

00912 

30 

tt        tt        tt 

11 

40 

40 

11 

43 

40 

65.4 

62.6 

ft 

43.25 

0.094G 

0.4125 

0.06316 

0.0957 

0.0963 

0.0960 

31 

"       25,   P.M. 

0 

15 

0 

0 

20 

0 

66.3 

62.8 

tf 

54.67 

0.1517 

0.4048 

0.12587 

0.1547 

0.1578 

0.1562 

32 

tt         tt        ti 

3 

27 

0 

3 

31 

30 

70.5 

63.6 

ft 

54.67 

0.1512 

0.4048 

0.12525 

0.1549 

0.1575 

0.1562 

33 

tt          t        tt 

3 

49 

10 

3 

53 

0 

70.7 

63.6 

tf 

44.00 

0.0998 

0.4118 

0.06833 

0.1013 

0.1017 

0.1015 

34 

ti          t        tt 

4 

10 

0 

4 

13 

30 

70.5 

63.7 

tt 

44.75 

0.1031 

0.4113 

0.07166 

0.1050 

0.1057 

0.1053 

35 

it                  t               tf 

4 

31 

0 

4 

34 

10 

70.5 

63.7 

it 

45.50 

0.1080 

0.4106 

0.07669 

0.1100 

0.1105 

0.1102 

36 

ti                  t               tt 

4 

57 

30 

5 

3 

30 

70.7 

(•  O    Q 

fi 

47.00 

0.1155 

0.4096 

0.08461 

0.1176 

01  •)  V  4 

0.1184 

01  OAA 

0.1180 
01  ono 

38 

tt                  t               tt 

5 
5 

20 
41 

0 
30 

5 
5 

23 
47 

0 
0 

70.0 
70.2 

o4.y 
64.0 

if 

50.00 
54.67 

0.1260 
0.1507 

0.4081 
0.4049 

0.09606 
0.12466 

.12o4 
0.1534 

.loOO 
0.1572 

.1  2u£ 
0.1553 

39 

"     26,  P.M. 

~2 

32 

10 

2 

35 

~30 

74.7 

64.6 

ABC 

60.00 

0.1775 

0.4023 

0.15833 

0.1889 

0.1897 

0.1093 

40 

tt      t..      tt 

3 

29 

0 

3 

32 

0 

75.0 

64.6 

tf 

32.50 

0.0292 

0.4215 

0.01107 

0.0297 

0.0296 

0.0296 

41 

tt      tt      tt 

3 

38 

30 

3 

41 

40 

75.1 

64.6 

tf 

34.25 

0.0396 

0.4201 

0.01742 

0.0402 

0.0401 

0.0401 

42 

tt      tt      tt 

3 

52 

30 

3 

56 

25 

75.1 

64.6 

ft 

36.50 

00557 

0.4179 

0.02891 

0.0566 

0.0567 

0.0566 

43 

tt      tt      it 

4 

5 

0 

4 

10 

40 

75.4 

64.6 

it 

38.50 

0.0677 

0.4162 

0.03858 

0.0694 

0.0690 

0.0692 

44 

ft             tt           -it 

4 

28 

40 

4 

32 

40 

75.5 

64.7 

tt 

40.50 

0.0795 

0.4146 

0.04891 

0.0814 

0.0812 

0.0813 

45 

tt        it        it 

4 

44 

10 

4 

46 

50 

76.0 

64.8 

tf 

42.50 

0.0914 

0.4129 

0.06004 

0.0939 

0.0940 

0.0939 

46 

it         it        tt 

4 

59 

30 

5 

3 

20 

76.6 

64.9 

ft 

45.50 

0.1067 

0.4108 

0.07535 

0.1103 

0.1103 

0.1103 

47 

tf        tt        ft 

5 

17 

40 

5 

22 

40 

75.6 

65.0 

tf 

50.00 

0.1271 

0.4080 

0.09729 

0.1321 

0.1319 

0.1320 

48 

it         tt        tt 

5 

41 

40 

5    46      0 

tf 

54.67 

0.1458 

0.4054 

0.11878 

0.1529 

0.1529 

0.1529 

49 

tt         tt        tt 

6 

10 

30 

6    18    50 

tt 

0.1696 

0.4031 

0.14817 

0.1804 

0.1808 

0.1806 

50 

Oct.    7,    A.M. 

9 

16 

30 

9    19      0 

58.2 

59.5 

it 

60.00 

0.1778 

0.4023 

0.15873 

0.1897 

0.1908 

0.1902 

XXVII. 

THROUGH  SUBMERGED  TUBES  AND   ORIFICES. 


219 


10 

11 

13 

13 

14 

15 

16 

17 

18 

Mean 

Effective 

Velocity 

Mean  ve- 

Ratio of 

Diameter 

Mean  ve- 

Ratio of 

height  of 

head  pro- 

due the 

locity  by 

the  veloci- 

of the 

locity  by 

the  veloci- 

the sur- 

ducing 

head  in 

experi- 

ty at  the 

tube  or 

experi- 

ty at  the 

No. 

face  of  the 

the  dis- 

the pre- 

ment 

smallest 

orifice  at 

ment  at 

final  dis- 

water in 

charge. 

ceding 

through 

section  to 

the  place 

the  final 

charge  to 

of 

compart- 

column. 

the  small- 

the veloci- 

of final 

discharge 

the  veloci- 

Remarks. 

ment  J?, 

est  section 

ty  due  the 

discharge. 

from  the 

ty  due  the 

the 

figures  1 

of  the 

head. 

tube. 

head. 

and  2, 

tube  or 

Exp. 

plato  XX, 

orifice. 

by  gauge 
D. 

H 

V 

V 

ti 

«' 

V' 

T 

Feet. 

Feet. 

Feet  per 
second. 

Feet  per 
second. 

Feet. 

Feet  per 
second. 

1 

0.0608 

0.0339 

1.4767 

1.2035 

0.8150 

0.1018 

1.2035 

0.8150 

On  the  completion  of  experiment  7,  the  water  was  drawn  out 

2 

0.0609 

0.0340 

1.4789 

1.2103 

0.8183 

tt 

1.2103 

0.8183 

of  the  cistern,  and   the  interior  of  the  mouth-piece  examined. 
Only  slight  traces  of  oxidation  were  observed.     In  order  to  pre- 

s 

0.1389 

0.0998 

2.5337 

2.0765 

0.8195 

tt 

2.0765 

0.8195 

vent  oxidation  before  the  experiments  were  resumed,  the  inte- 

4 
5 

0.1384 
0.2117 

0.1002 
0.1647 

2.5387 
3.2549 

2.0369 
2.7347 

0.8024 
0.8402 

a 
tt 

2.0369 
2.7347 

0.8024 
0.8402 

rior  was  wiped  dry,  and  smeared  with  a  grease  consisting  of 
about  20  parts  of  beef  tallow,  10  parts  of  fine  sperm  oil,  and  1 
part  of  beeswax.     The  cistern  remained  empty  until  the  experi- 

6 

0.2835 

0.2300 

3.8464 

3.3180 

0.8626 

tt 

3.3180 

0.8626 

ments  were  resumed,  September  21st,  when,  previous  to  experi- 
ment 8,  the  grease  was  removed  by  thoroughly  rubbing   the 

7 

0.3790 

0.3178 

4.5213 

4.0341 

0.8923 

tt 

4.0341 

0.8923 

surface  with  cloth  and  cotton-waste. 

8 

0.3735 

o.siis 

4.4784 

4.1222 

0.9205 

tt 

4.1222 

0.9205 

Experiment  8  was  a  repetition  of  experiment  7  ;  the  increased 
discharge  observed  in  experiment  8  must  be  attributed  to  the 

9 

0.4838 

0.4154 

5.1691 

4.7719 

0.9232 

u 

4.7719 

0.9232 

change  in  the  state  of  the  surface,  due  to  the  greasing  and  wip- 

10 

0.6010 

0.5269 

5.8217 

5.3945 

0.9266 

tt 

5.3945 

0.9266 

ing  previously  described. 
At6h  30™  P.M.,  September  21st,  the  water  wasdrawn  outof  the 

11 

12 

0.7390 
0.8616 

0.6590 
0.7767 

6.5107 
7.0682 

6.0985 
6.6070 

0.9367 
0.9348 

tt 
tt 

6.0985 
6.6070 

0.9367 
0.9348 

cistern  and  the  interior  of  the  mouth-piece  examined.     A  large 
part  of  the  surface  at  and  near  the  smallest  section,  where  the 
velocity  of  the   water   was  greatest,   was  covered   with   oxida- 

13 

.0486 

0.9579 

7.8495 

7.2822 

0.9277 

tt 

7.2822 

0.9277 

tion  ;  this  was  rubbed  off  with  a  cloth,  when  the  previous  lus- 

14 

.1782 

1.0833 

8.3475 

7.7481 

0.9282 

u 

7.7481 

0.9282 

tre  of  the  surface  was  observed  to  be  tarnished.     It  was  then 
greased  anew.     The  water  was  left  out  of  the  cistern  until  the 

15 

.3322 

1.2327 

8.9046 

8.3065 

0.9328 

tt 

8.3065 

0.9328 

experiments  were  resumed   September  22d,  A.M.,   previous   to 

16 

.5008 

1.3968 

9.4788 

8.8786 

0.9367 

tt 

8.8786 

0.9367 

which  the  grease  was  wiped  off.    Experiment  18  was  a  repetition 
of  experiment  17,  for  the  purpose  of  ascertaining  the  effect  of 

17 
18 

.6214 
.6232 

1.5140 
1.5158 

9.8684 
9.8743 

9.2837 
9.3205 

0.9407 
0.9439 

tt 
tt 

9.2837 
9  3205 

0.9407 
0.9439 

the  change  in  the  state  of  the  surface.     There  was  no  change  in 
the  discharge,  however,  that  could  be  attributed  to  the  change 
in  the  state  of  the  surface. 

19 

1.6235 

1.4690 

9.7207 

15.2853 

1.5725 

0.1454 

7.4928 

0.7708 

After  the  conclusion  of  the  experiments  Septeinlmr  22d,  the 

20 

0.0485 

0.0202 

1.1399 

1.3116 

1.1506 

tt 

0.6429 

0.5640 

water  was  drawn  out  of  the  cistern  and  the  mouth-piece  and  the 
first  joint  of  the  diverging  tube  were  greased.     The  cistern  re- 

21 
22 

0.0873 
0.1204 

0.0481 

0.0727 

1.7590 
2.1625 

2.1162 

2.7781 

1.2031 
1.2847 

u 

u 

1.0374 
1.3618 

0.5897 
0.6297 

mained  empty  until  9  A.M.,  September  24th,  when  it  was  filled. 
September  25th,  A.M.,  previous  to  the  commencement  of  the  ex- 
periments, the  cistern  was  emptied  and  the  grease  wiped  off  the 

23 

0.1552 

0.0996 

2.5311 

3.5329 

1.3958 

it 

1.7318 

0.6842 

interior  of  the  mouth-piece  and  first  joint  of  the  diverging  tube. 

24 

0.1923 

0.1302 

2.8939 

4.1423 

1.4314 

tt 

2.0305 

0.7017 

25 

0.2327 

0.1643 

3.2509 

4.7508 

1.4614 

u 

2.3288 

0.7164 

26 

0.2745 

0.2005 

3.5912 

5.3193 

1.4812 

tt 

2.6075 

0.7261 

At  2h  25m  P.M.,  September  25th,  the  cistern  was  emptied  and 

27 

0.3286 

0.2482 

3.9956 

6.0203 

1.5067 

tt 

2.9511 

0.73o6 

the  interior  of  the  pipes  examined.     The  mouth-piece  was  free 
from  oxidation,  the  first  joint  of  the  diverging  tube  was  oxidated 

28 

0.3836 

0.2974 

4.3738 

6.6187 

1.5133 

tt 

3.2445 

0.7418 

sufficiently  to  redden  the  fingers  when  rubbetl  upon  it;  both  the 

29 
30 

0.4370 
0.4920 

0.3458 
0.3960 

4.7163 
5.0470 

7.2238 
7.7604 

1.5317 
1.5376 

tt 
tt 

3.5410 
3.8041 

0.7508 
0.7537 

pipes  were  wiped  clean  and  dry,  then  coated  with  grease  which 
was  afterwards  wiped  off  as  much  as  practicable  by  rubbing  with 
a  cloth.     Experiment  32  was  a  repetition  of  31,  to  ascertain  the 

31 

1.6179 

1.4617 

9.6965 

15.4647 

1.5949 

a 

7.5807 

0.7818 

effect  due  to  the  state  of  the  surface  caused  by  cleaning  and 
greasing.     The  change  in  the  discharge,  however,  due  to  this 

32 

1.6023 

1.4461 

9.6446 

15.3883 

1.5955 

tt 

7.5432 

0.7821 

cause,  was,  if  any,  extremely  small. 

33 

0.5470 

0.4455 

5.3531 

8.3947 

1.5682 

tt 

4.1150 

0.7687 

After  the  conclusion  of  the  experiments  September  25th,  P.M. 
the  cistern  was  emptied;  the  mouth-piece  WHS  found  free  from 

34 

0.5971 

0.4918 

5.6244 

8.8038 

1.5653 

tt 

4.3155 

0.7673 

oxidation,  and  the  first  joint  of  the  diverging  pipe  was  only 

35 

0.6660 

0.5558 

5.9792 

9.4227 

1.5759 

tt 

4.6190 

0.7725 

slightly  oxidated;  both  pipes  were  greased  and  tho  cistern  filled 
with  water. 

36 

0.7850 

0.6670 

6.5501 

10.3957 

1.5871 

tt 

5.0959 

0.7780 

37 

0.9836 

0.8544 

7.4134 

11.8017 

1.5919 

a 

5.7851 

0.7804 

38 

1.6257 

1.4704 

9.7253 

15.3158 

1.5748 

tt 

7.5077 

0.7720 

39 
40 

1.6040 
0.0439 

1.4147 
0.0143 

9.5393 
0.9591 

19.4523 
1.3599 

2.0392 
1.4179 

0.2339 

tt 

3.6847 
0.2576 

0.3863 
0.2686 

September  26  I1'   25m   P.M.    The  cistern  lias  stood  full  of  water 
since  last  evening  i  the  water  was  now  drawn  off,  and  the  grmse 
wiped   off  the   mouth-piece   and  first  joint  of  diverging  pipe. 

41 

0.0710 

0.0309 

1.4098 

2.1405 

1.5183 

ti 

0.4055 

0.2876 

The  second  joint  was  then  put  on  for  the  experiments  of  to-day. 

42 

0.1182 

0.0616 

1.9906 

3.5520 

1.7844 

a 

0.6728 

0.3380 

43 

0.1667 

0.0975 

2.5043 

4.7403 

1.8929 

« 

0.8979 

0.3586 

44 

0.2241 

0.1428 

3.0307 

6.0090 

1.9827 

'  (t 

1.1383 

0.3756 

October  7,  A.M.     The  cistern  has  been  kept  full  of  water  since 

45 

0.2993 

0.2054 

3.6348 

7.3771 

2.0296 

tt 

1.3974 

0.3845 

September  26th,  excepting  on  two  or  three  occasions,  when  it 
was  emptied,  to  permit  the  tubes  to  be  cleaned  and  greased  anew. 

46 

0.4220 

0.3117 

4.4777 

9.2576 

2.0675 

tt 

1.7536 

0.3916 

This  morning,  on  emptying  the  cistern  and  wiping  off  the  grease, 

47 

0.6271 

0.4951 

5.6427 

11.9537 

2.1184 

u 

2.2643 

0.4013 

no  oxidation  was  observed. 

48 

0.8673 

0.7144 

6.7788 

14.5929 

2.1527 

a 

2.7643 

0.4078 

49 

1.2805 

1.0999 

8.4113 

18.2043 

2.1643 

u 

3.4483 

0.4100 

50 

1.5018 

1.3116 

9.1851 

19.5016 

2.1232 

it 

3.6941 

0.4022 

220 


TABLE 
EXPERIMENTS  ON  THE  FLOW   OF  WATER 


1 

3 

3 

4 

5 

6 

7 

8 

9 

Time  of  making  the  obser- 
vations from   which  the 

Temperature, 
in  degrees 

Reference  to 
figure  2, 

Position 
of  the 

Mean 
depth  of 

Value  of 
C  in  the 

Quantity 
of  water 

Ileight  of  the  surface  of  the 
water  in  compartment  Ft 

No. 

Date. 

mean  heights  given 
in  this  table  are 

of  Fahren- 
heit's ther- 

plate XXI., 
indicating 

index  of 
the  in- 

water on 
the  weir. 

formula 
in  the 

discharged, 
calculated 

figures  1  and  2, 
plate  XX. 

of 

deduced. 

mometer  ; 

the  parts  of 

let  cock. 

by  gauge 

next  col- 

by the 

the  com- 

A. 

umn. 

formula 

the 

pound  tube 

1854. 

used. 

h 

D  = 

Exp. 

Beginning. 

Ending. 

G'l/il/  'Igk 

by  gauge 
B. 

by  gauge 
C. 

Mean. 

H. 

Mln. 

Sec. 

H. 

Min. 

Sec. 

of  the 
air. 

of  the 
water. 

Degrees. 

Feet. 

Cubic  feet 
per  second. 

Feet. 

Feet. 

Feet. 

51 

Oct.    7,    A.M. 

10 

59 

0 

11 

1 

30 

66.0 

60.5 

ABCD 

62.00 

0.1874 

0.4014 

0.17137 

0.2055 

0.2058 

0.2056 

52 

ti       it       tt 

11 

18 

0 

11 

20 

30 

66.1 

60.6 

it 

32.50 

0.0284 

0.4217 

0.01062 

0.0290 

0.0288 

0.0289 

53 

u        u       u 

11 

40 

0 

11 

42 

40 

66.1 

60.6 

If 

34.25 

0.0394 

0.4201 

0.01729 

0.0405 

0.0404 

0.0404 

54 

U            ((           It 

11 

51 

0 

11 

53 

40 

66.1 

60.6 

It 

36.50 

0.0555 

0.4179 

0.02876 

0.0575 

0.0574 

0.0574 

55 

"        7,    P.M. 

2 

16 

0 

2 

18 

30 

68.6 

59.8 

ff 

38.50 

0.0668 

0.4163 

0.03783 

0.0701 

0.0700 

0.0700 

56 

It          It          it 

2 

27 

0 

2 

29 

0 

69.0 

59.6 

It 

40.50 

0.0801 

0.4145 

0.04945 

0.0846 

0.0848 

0.0847 

57 

ff            It           (1 

2 

35 

40 

2 

29 

0 

69.1 

59.5 

tf 

42.50 

0.0908 

0.4130 

0.05947 

0.0963 

0.0962 

0.0962 

58 

(t         (t        (t 

2 

47 

10 

2 

51    30 

69.1 

59.4 

tt 

45.50 

0.1083 

0.4106 

0.07701 

0.1157 

0.1157 

0.1157 

-59 

ii        11        It 

3 

6 

0 

3 

11 

0 

69.5 

59.3 

ii 

50.00 

0.1273 

0.4079 

0.09750 

0.1372 

0.1372 

0.1372 

60 

(t        it       a 

3 

22 

0 

3 

26 

40 

69.9 

59.3 

It 

54.67 

0.1462 

0.4053 

0.11924 

0.1593 

0.1595 

0.1594 

61 

it           U           If 

3 

42 

20 

3 

47 

30 

70.1 

59.4 

ft 

60.00 

0.1700 

0.4030 

0.14866 

0.1875 

0.1880 

0.1877 

e-2 

it         ft         (i 

4 

17 

0 

4 

22 

30 

70.9 

59.5 

ft 

62.00 

0.1880 

0.4013 

0.17215 

0.2098 

0.2102 

0.2100 

63 

it           U           It 

4 

40 

10 

4 

45 

0 

71.3 

59.7 

II 

63.50 

0.197-1 

0.4004 

0.18481 

0.2215 

0.2216 

0.2215 

64 

Oct.    10,  A.M. 

8 

43 

0 

8 

47 

0 

61.2 

59.0 

ft 

62.00 

0.1895 

0.4012 

0.17417 

0.2063 

0.2066 

(1.2(1114 

~65 

ft           tl           U 

9 

51 

30 

3 

~55 

30 

65.0 

59.2 

ABCDE 

62.50 

0.1907 

0.4010 

0.17574 

0.2100 

0.2101 

0.2100 

66 

11          U          tt 

10 

55 

30 

11 

5 

30 

63.8 

59.0 

tf 

62.50 

0.1893 

0.4012 

0.17390 

0.2091 

0.2094 

0.2092 

67 

ft          11          tt 

11 

17 

30 

11 

22 

30 

64.0 

59.0 

If 

32.50 

0.0292 

0.4215 

0.01107 

0.0300 

0.0298 

0.0299 

68 

tt          (f          tt 

11 

44 

0 

11 

47 

0 

41 

34.25 

0.0390 

0.4202 

0.01703 

0.0404 

0.0401 

0.0402 

69 

"         "      P.M. 

2 

4 

30 

2 

8 

30 

64.3 

59.1 

tt 

35.50 

0.0460 

0.4192 

0.02177 

0.0481 

0.0478 

0.0479 

70 

tt        tt        tt 

2 

23 

30 

2 

28 

30 

64.8 

59.3 

tf 

36.50 

O.C563 

0.4178 

0.02937 

0,0589 

0.0587 

0.0588 

71 

tf          ft          U 

2 

43 

30 

o 

46 

30 

ft 

37.50 

0.0621 

0.4170 

0.03396 

0.0652 

0.0649 

0.0650 

72 

tt        tt       tt 

2 

58 

0 

3 

2 

30 

65.0 

59.5 

II 

38.50 

0.0680 

0.4162 

0.03884 

0.0716 

0.0712 

0.0714 

73 

tt        tt       u 

3 

33 

30 

3 

38 

0 

65.3 

59.6 

tt 

39.50 

0.0745 

0.4153 

0.04444 

0.0788 

0.0785 

0.0786 

74 

tt        tt       ti 

3 

51 

30 

3 

57 

30 

65.6 

59.7 

It 

40.50 

0.0801 

0.4145 

0.04945 

0.0849 

0.0847 

0.0848 

75 

ft        tt       tt 

4 

14 

0 

4 

21 

0 

66.1 

59.7 

ff 

41.50 

0.0848 

0.4138 

0.05378 

0.0901 

0.0897 

0.0899 

76 

it        tt        tt 

4 

34 

0 

4 

40 

0 

66.5 

59.8 

ft 

42.50 

0.0916 

0.4129 

0.06024 

0.0978 

0.0975 

0.0976 

77 

tt        tt       tt 

4 

57 

0 

5 

1 

0 

66.2 

59.8 

If 

43.25 

0.0960 

0.4123 

0.06454 

0.1025 

0.1023 

0.1024 

78 

it        tt       tt 

5 

40 

0 

5 

42 

0 

ff 

62.50 

0.1931 

0.4008 

0.17898 

0.2191 

0.2184 

0.2187 

79 

"       12.A.M. 

8 

33 

0 

8 

37 

30 

62.8 

59.5 

ii 

62.50 

0.1906 

0.4011 

0.17565 

0.2092 

0.2090 

0.2091 

80 

it               if               it 

8 

51 

30 

8 

56 

30 

62.7 

59.5 

11 

44.00 

0.1003 

0.4117 

0.06882 

0.1041 

0.1042 

0.1041 

81 

ft               ft              ft 

9 

9 

30 

9 

17 

30 

62.8 

59.6 

(t 

44.75 

0.1042 

0.4111 

0.07277 

0.1090 

0.1087 

0.1088 

82 

it               11              ft 

9 

29 

0 

9 

35 

0 

63.1 

59.6 

ft 

45.50 

0.1128 

0.4099 

0.08172 

0.1189 

0.1184 

0.1186 

83 

it               (t              if 

9 

51 

30 

9 

57 

30 

63.2 

59.6 

ft 

47.00 

0.1150 

0.4096 

0.08406 

0.1210 

0.1206 

0.1208 

84 

tt               ft              tt 

10 

12 

0 

10 

17 

0 

64.0 

59.6 

tf 

50.00 

0.1275 

0.4079 

0.09773 

0.1348 

0.1346 

0.1347 

85 

tt               ft              ft 

10 

38 

30 

10 

44 

0 

65.0 

59.7 

ft 

54.67 

0.1471 

0.4052 

0.12031 

0.1575 

0.1575 

0.1575 

86 

tf               ft              it 

11 

6 

30 

11 

10 

0 

65.6 

59.8 

tl 

60.00 

0.1697 

0.4031 

0.14830 

0.1846 

0.1850 

0.1848 

87 

tf               ft              tf 

11 

34 

30 

11 

37 

30 

67.6 

59.9 

ft 

62.00 

0.1896 

0.4012 

0.17431 

0.2095 

0.2088 

0.2091 

88 

tf               tt              tt 

11 

53 

30 

11 

58 

30 

tf 

62.50 

0.1911 

0.4010 

0.17630 

0.2114 

0.2117 

0.2115 

89 

"        "      P.M. 

2 

40 

0 

2 

50 

0 

69.8 

60.3 

U 

62.50 

0.1917 

0.4010 

0.17713 

0.2131 

0.2136 

0.2133 

90 

ft        ft        tt 

3 

3 

0 

3 

11 

30 

69.8 

60.3 

II 

62.50 

0.1919 

0.4009 

0.17736 

0.2136 

0.2143 

0.2139 

91 

ft        tt        it 

4 

20 

0 

4 

25 

0 

69.3 

60.6 

A 

45.50 

(1.1(177 

0.4107 

0.07639 

0.1124 

0.1144 

0.1134 

92 

it             U            ii 

5 

27 

30 

5 

30 

30 

~7h6 

•60.7 

ABCDE 

62.50 

0.1917 

0.4010 

0.17713 

0.2204 

0.2209 

0.2206 

93 
94 
95 

"       16,  A.M. 

tt         ti        ii 

ft         tt        it 

9 
9 
11 

18 
58 
4 

30 
0 
0 

9 
10 
11 

21 
1 

7 

30 
30 
30 

55.0 
56.1 
56.6 

57.1 
57.1 
57.6 

40.00 
32.50 
35.50 

U.077S 
0.0293 
0.0494 

0.4148 
0.4215 
0.4187 

0.04737 
0.01113 
0.02419 

0.0786 
0.0299 
0.0504 

0.0793 
0.0296 
0.0503 

0.0789 
0.0297 
0.0503 

96 

tt         tt        tt 

11 

39 

0 

11 

45 

80 

59.2 

58.0 

• 

36.50 

0.0522 

0.4184 

0.02626 

0.0541 

0.0533 

0.0537 

97 

"         u      P.M. 

2 

10 

0 

2 

14 

0 

40.00 

0.0777    0.4148 

0.04728 

0.0796 

0.0798 

0.0797 

98 

ft            tl           U 

2 

40 

0 

2 

44 

0 

65.2 

37.50 

0.0618    0.4170 

0.03372 

0.0634 

0.0639 

0.0636 

99 

ft             U             It 

2 

59 

0 

3 

3 

0 

65.6 

57.5 

38.50    0.0682    0.4161 

0.03900 

0.0703 

0.0702 

0.0702 

100 

tf         ff        II 

3 

l!i 

0 

s 

23 

0 

65.7 

57.6 

39.50 

0.0744 

0.4153 

0.04435 

0.0769 

0.0769 

0.0769 

101 

(t       If       II 

•    4 

11 

0 

4 

14 

0 

61.9 

57.8 

40.00 

0.0775 

0.4148    0.04710 

0.071)9 

0.0804 

0.0801 

XXVII  —  CONTINUED. 

THROUGH  SUBMERGED  TUBES  AND   ORIFICES. 


221 


1O 

11 

12 

13 

14 

15 

16 

17 

18 

Mean 

Effective 

Velocity 

Mean  ve- 

Ratio of 

Diameter 

Mean  ve- 

Ratio of 

height  of 

head  pro- 

due the 

locity  by 

the  veloci 

of  the 

locity  by 

the  veloci 

the  sur- 

ducing 

head  in 

experi- 

ty at  the 

tube  or 

experi- 

ty at  the 

No. 

face  of  the 

the  dis- 

the pre- 

ment 

smallest 

orifice  at 

ment  at 

final  dis- 

water in 

charge. 

ceding 

through 

section  to 

the  place 

the  final 

charge  to 

of 

compart- 

column. 

the  small 

the  veloci 

of  final 

discharge 

the  veloci 

ment  E, 

est  section 

ty  due  the 

discharge 

from  the 

ty  due  the 

Remarks. 

the 

figures  1 

of  the 

head. 

tube. 

head. 

and  2, 

tube  or 

Exp 

plate  XX 

orifice. 

by  gauge 
D. 

H 

7 

V 

V 

r 

«' 

V 

T  • 

Feet 

Feet. 

Feet  per 

Feet  per 

Feet  per 

second. 

scrnml. 

ee  . 

second. 

51 

1.6327 

1.4271 

9.5810 

21.055C 

2.1976 

0.3209 

2.1189 

0.2212 

At  8h  35m  A.M.,  October  7,  the  diaphragm  of  strainer  cloth  in 

52 

0.0427 

0.0138 

0.9422 

1.3050 

1.3850 

0.1313 

0.1394 

the  gauging  basin  was  cleaned  ;  it  had  become  obstructed  by  an 
accumulation  of  gummy  matter,  apparently  an  exudation  from 

53 

0.0809 

0.0405 

1.6140 

2.1243 

1.3162 

ti 

0.2138 

0.1325 

the  new  pine  planks  of  which  the  cistern  was  constructed. 

54 
55 

0.1162 
0.1581 

0.0588 
0.0881 

1.9448 
2.3805 

3.5329 
4.6472 

1.8166 
1.9522 

ti 

0.3555 
0.4677 

0.1828 
0.1965 

October  7,  P.M.     After  the  conclusion  of  experiment  (33,  the 
cistern  was  emptied  and  the  three  joints  B,  0,  and  D  of  the  di- 
verging tube  taken  off  and  examined  ;  all  of  them,  together  with 

56 

0.2173 

0.1326 

2.9205 

6.0757 

2.0804 

u 

0.6114 

0.2094 

the  mouth-piece,  were  a  little  oxidated,  the  mouth-piece  the  least 
so,  and  the  joints  C  and  D  the  most  ;  they  were  then  all  wiped 

57 

0.2773 

0.1811 

3.4131 

7.3064 

2.1407 

it 

0.7353 

0.2154 

clean  and  coated  anew  with  grease  ;  the  diverging  tube  was  not 

58 

0.3901 

0.2744 

4.2012 

9.4620 

2.2522 

ti 

0.9522 

0.2267 

put  on  again  to-day. 
October  10,  A.M.    The  cistern  has  been  kept  full  of  water 

59 

0.5740 

0.4368 

5.3006 

11.9790 

2.2599 

ti 

1.2055 

0.2274 

since  October  7.     This  morning  it  was  emptied,  and  the  grease 

60 

0.7887 

0.6293 

6.3623 

14.6494 

2.3025 

n 

1.4743 

0.2317 

wiped  off  the  mouth-piece  ;  the  joints  B,  C,  and  D  were  put  on, 
the  grease  having  been  first  wiped  off. 

61 

1.1048 

0.9171 

7.6806 

18.2642 

2.3780 

it 

1.8380 

0.2393 

62 

1.3872 

1.1772 

8.7018 

21.1509 

2.4306 

n 

2.1286 

0.2446 

63 

1.5827 

1.3612 

9.3572 

22.7058 

2.4266 

u 

2.2850 

0.2442 

64 

1.5952 

1.3888 

9.4516 

21.3992 

2.2641 

** 

21535 

0.2278 

65 

1.6283 

1.4183 

9.5514 

21.5920 

2.2606 

0.4085 

1.3409 

0.1404 

At  9k  0    October  1(1,  the  cistern  was  emptied  and  the  joint  E 

put  on- 

66 

1.6165 

1.4073 

9.5143 

21.3653 

2.2456 

ti 

1.3268 

0.1395 

No  change  was  made  in  the  apparatus  between  experiments 

67 
68 

0.0438 
0.0687 

0.0139 
0.0285 

0.9456 
1.3540 

1.3599 
2.0925 

1.4381 
1.5455 

u 
ti 

0.0845 
0.1300 

0.0893 
0.0960 

65  and  66;    the  water  flowed  continuously  from  9h  20TO  until 
after  the  conclusion  of  experiment  66. 
October  10,  P.M.    After  the  conclusion  of  experiment  78  the 

69 
70 

0.0858 
0.1163 

0.0379 
0.0575 

1.5614 
1.9232 

2.6741 
3.6087 

1.7126 
1.8764 

ti 

n 

0.1661 
0.2241 

0.1064 
0.1165 

cistern  was  emptied,  and  the  four  joints  of  the  diverging  tube 
ta&en  off.     There  were  only  a  few  slight  streaks  of  oxidation  on 
the  mouth-piece  ;  the  joints  B  and  C  of  the  diverging  tube  were 

71 

0.1374 

0.0724 

2.1580 

4.1725 

1.9335 

n 

0.2591 

0.1201 

oxidated  in  longitudinal  streaks  ;  joints  D  and  E  were  nearly  cov- 
ered with  oxidation,  which  was  however  rubbed  off  with  ease, 

72 

0.1596 

0.0882 

2.3819 

4.7719 

2.0034 

ti 

0.2963 

0.1244 

leaving  the  surface,  apparently,  as  smooth  as  before.     The  inte- 

73 
74 

0.1884 
0.2163 

0.1098 
0.1315 

2.6576 
2.9084 

5.4603 
6.0757 

2.0546 
2.0890 

tt 

0.3391 
0.3773 

0.1276 
0.1297 

rior  of  the  mouth-piece  and  of  the  four  joints  of  the  diverging 
tube   were  wiped  clean  and  coated  with  grease  ;  the  diverging 
tube  was  not  put  on  again  to-day. 

75 

0.2423 

0.1524 

3.1310 

6.6070 

2.1102 

it 

0.4103 

0.1310 

October  12,  A.M.     The  apparatus  was  prepared  for  the  experi- 
ments of  to-day  by  removing  the  grease  from  the  interior  of  the 

76 

0.2848 

0.1872 

3.4701 

7.4013 

2.1329 

it 

0.4596 

0.1325 

mouth-piece  and  four  joints  of  the  diverging  tube,  and  putting 

77 

0.3104 

0.2080 

3.6578 

7.9294 

2.1678 

n 

0.4924 

0.1346 

the  latter  in  their  places. 
At  3h  15"  P.M.,  October  12,  the  cistern  was  emptied  and  the 

78 

1.5010 

1.2823 

9.0820 

21.9899 

2.4213 

it 

1.3656 

0.1504 

tube  examined  ;  the  interior  of  the  mouth-piece  and  all  the 

711 
80 

1.6176 
0.3261 

1.4085 
0.2220 

9.5184 
3.7789 

21.5804 
8.4558 

2.2672 
2.2376 

it 
ti 

1.3402 
0.5251 

0.1408 
0.1390 

joints  were  oxidated,  and  in  a  little  greater  degree  than  after 
experiment  78  as  noted  above.     The  four  joints  of  the  diverging 
tube  were  taken  off,  and  together  with  the  mouth-piece  were  well 

81 

0.3539 

0.2451 

3.9706 

8.9407 

2.2517 

u 

0.5552 

0.1398 

rubbed  with  a  cloth,  which  removed  all  the  red  oxide. 

82 

0.4248 

0.3062 

4.4380 

10.0407 

2.2624 

tl 

0.6236 

0.1405 

83 

0.4397 

0.3189 

4.5291 

10.3283 

2.2804 

It 

0.6414 

0.1416 

84 

0.5557 

0.4210 

5.2039 

12.0072 

2.3073 

11 

0.7457 

0.1433 

85 

0.7987 

0.6412 

6.4222 

14.7812 

2.3016 

tl 

0.9180 

0.1429 

86 

1.1483 

0.9635 

7.8725 

18.2204 

2.3144 

11 

1.1315 

0.1437 

87 

1.5575 

1.3484 

9.3131 

21.4161 

2.2996 

tt 

1.3300 

0.1428 

88 

1.5884 

1.3769 

9.4110 

21.6600 

2.3016 

tl 

1.3451 

0.1429 

89 

1.5745 

1.3612 

9.3572 

21.7621 

2.3257 

11 

1.3515 

0.1444 

90 

1.5588 

1.3449 

9.3010 

21.7907 

2.3428 

It 

1.3533 

0.1455 

91 

1.6285 

1.5151 

9.8720 

9.3858 

0.9507 

0.1018 

9.3858 

0.9507 

92 

1.5069 

1.2863 

9.0961 

21.7621 

2.3925 

0.4085 

1.3515 

0.1486 

•  At  4J>  30™  P.M.,  October  12,  the  cistern  was  emptied  again, 
and  the  four  joints  of  the  diverging  tube  re-attached.    At  6  P.M. 
the  cistern  was  emptied  ;  the  wide  part  of  the  mouth-piece  was 

much  oxidated,  but  only  slightly  so  at  the  smallest  section. 

The  diverging  tube  was  oxidated  in  only  a  few  spots. 

93 
94 

1.5925 
0.1213 

1.5136 
0.0916 

9.8671 
2.4274 

5.8316 
1.3695 

0.5910 
0.5642 

0.1017 

It 

5.8316 
1.3695 

0.5910 
0.5642 

Orifice  in  a  thin  plate. 
The  plate,  Figs.  12  and  13,  plate  XXI.,  containing  the  orifice, 
was  put  on  October  14  ;  the  accessible  parts  of  it  were  greased, 

95 

0.4855 

0.4352 

5.2909 

2.9783 

0.5629 

11 

2.9783 

0.5629 

and  the  cistern  filled  with  water,  and  so  remained  until  October 

16,  A.M.,  when  it  was  emptied,  and  the  grease  wiped  off.     No 

96 

0.5372 

0.4835 

5.5768 

3.2328 

0.5797 

It 

3.2328 

0.5797 

oxidation  was  observed. 

97 
98 

1.5784 
0.8400 

1.4987 
0.7764 

9.8184 
7.0G69 

5.8203 
4.1504 

0.5928 
0.5873 

tt 

5.8203 
4.1504 

0.5928 
0.5873 

At  O1*  15"  P.M.,  October  16,  the  cistern  was  emptied  and  the 
alate  examined  ;  there  was  a  thin  coating  of  oxide  over  most  of 
the  surface  ;  all   the  accessible  parts  of  tbe  plate  were  wiped 

99 

1.0944 

1.0242 

8.1167 

4.8012 

0.5915 

It 

4.8012 

0.5915 

clean  and  greased  anew.     At  lh  15'°  P.M.,  the  grease  was  wiped 

100 

1.4004     1.3235 

9.2267 

5.4601 

0.5918 

It 

5.4601 

0.5918 

off  again. 

101 

1.5704     1.-I003 

9.7909 

5.7979 

0.5922 

It 

5.7979 

0.5922 

222. 


EXPERIMENTS  ON  THE  FLOW  OF  WATER 


DESCRIPTION  OF  TABLE  XXVIL,   CONTAINING  THE  EXPERIMENTS  ON  THE  FLOW   OF 
WATER  THROUGH  SUBMERGED  TUBES  AND  ORIFICES. 

258.  The  greater  portion  of  this  table  will  be  intelligible  from  the  headings 
of  the  several  columns,  without  further  explanation. 

As  previously  stated,  the  quantity  of  water  flowing  was  gauged  by  means  of 
a  weir  of  substantially  the  same  form  and  dimensions  as  that  used  by  Fencelet 
and  Lesbros,  in  their  experiments  made  at  Metz  in  1827  and  1828.  Table  X., 
Experiences  hydrauliques,  &c.,  previously  cited,  contains  the  results,  of  the  exper- 
iments made  in  1828.  The  quantities  E  discharged  by  experiment  with  certain 
depths  on  the  weir  are  given  ;  also  the  quantities  with  the  same  depths,  com- 

.  _  tr 

puted  by  the  formula  d=lh^2gh;   also  the  values  of  -j.      These   last   quantities 

are   the   values   of  the   coefficient  C,   by  means  of  which   the  real  discharge   can   be 

deduced  from   the   value    of  d.      We   can   then  compute   the   real   discharge   by    the 
formula 


The  value  of  C  is  not  the  same  for  all  depths,  as  may  be  seen  by  the  follow- 
ing table,  which  contains  the  principal  results  of  table  X.  of  Poncelet  and  Lesbros 
above  cited,  changing  the  unit  from  metres  to  English  feet.  The  length  of  the 
weir  I  was  0.10  metres  or  0.6562  foot. 


Depth  of  water  on 
the  weir,  taken  11.48 
feet  up  stream  from 
the  weir. 

h 

Discharge  by 

experiment. 

E 

Discharge  computed 
by  the  formula 

Value  of  C  in  the 
formula. 

d=lh^2gh. 

D=Clh)/2gh, 

Feet. 

Cubic  feet  per  second. 

Cubic  feet  per  second. 

0.6821 

1.1528 

2.9656 

0.3888 

0.5351 

0.8098 

20608 

0.3930 

0.3376 

0.4071 

•  1.0327 

0.3943 

0.1985 

0.1864 

0.4655 

0.4003 

0.1463 

0.1194 

0.2947 

0.4053 

0.0771 

0.0468 

0.1127 

0.4149 

The  values  of  C,  given  in  column  7,  are  deduced  from  the  values  of  C  in 
the  preceding  table,  by  interpolation.  The  quantities  of  water  discharged  by  the 
tube  or  orifice  given  in  column  8  are  computed  by  the  formula  D  =  C  I  h  y/  2  c  h, 
in  which  C  has  the  value  given  in  column  7;  the  length  of  the  weir  I,  by 


THROUGH   SUBMERGED   ORIFICES   AND   DIVERGING   TUBES.  223 

measurement,  =  0.6579  foot;  h  =  the  value   given   in   column  6,  and  g  =  32.1618, 
which  is  its  value  for  the  place  where  the  experiments  were  made  (art.  68). 

259.  As   previously   stated,   according   to    the   first   design  of  the  apparatus,  the 
weir   was   intended    to   be   placed   in   the   partition   JV,   figures    1    and   2,   plate    XX., 
and   the   depth   on   the   weir  was   intended   to   be   measured  by  the  hook  gauge  B ; 
on   trial,  however,  it  was   found   that  the    agitation   in   the    compartment   F  was  too 
great   to   admit   of  a   satisfactory  gauge   being  made  with   the  weir   in    this  position, 
and   it   was   accordingly   removed   to   the    position   represented   in   the    figures.      The 
hook  gauge   B   was   allowed   to  remain,  and   the  height  of  the  surface  of  the  water 
in  the   compartment  F  was   observed   by  means  of  both   the   gauges  B  and  C,  and 
the   mean   of   the   two   is   taken   as    the   elevation   of   the   surface   of   the   water  in 
this   compartment.      By    comparing   the   heights  taken   at  the   two   gauges,   given   in 
column    9,   it  will   be    seen   that,  when  the   quantity  of  water   discharged  was  small, 
there    was   little    or   no    difference   in   the    indications   of  the   two   gauges;    with   the 
larger  volumes,  the  height  at  gauge  B  was  sensibly  the  greatest. 

The    effective    head    producing    the    discharge    given   in   column    11    is    the    dif- 
ference of  the  heights  of  the  surface  of  the  water  in  compartments  E  and  F. 
The  velocity  given  in  column  12  is  computed  by  the  formula    V  =.  \l  2  g  h. 

260.  The    smallest    section    of    the    compound   tube   is   in   the   mouth-piece   be- 
tween  a   and    6,   figure    2,   plate    XXI.,    and    was    found,    by    careful    and    repeated 
measurements  made   by  different  persons,   to   be  0.1018   foot.     The  diameter  of   the 
orifice   in   the   thin   plate   was   found   in   a   similar   manner   to   be    0.1017   foot.     The 
area   of  the    orifice  in  the   mouth-piece  was  consequently  0.0081393  square   foot,  and 
the  area  of  the   orifice  in  the  thin  plate  was  0.0081233  square  foot.     The  velocities 
given   in    column    13    are    obtained    by   dividing   the    quantities    given   in   column   8 
by  the  area  of  the  smallest  section  through  which  the  water  was  discharged. 


DEDUCTIONS  FROM  THE  EXPERIMENTS  GIVEN  IN  TABLE  XXVII. 

261.  Confining  ourselves,  for  the  present,  to  the  velocities  at  the  smallest 
section,  we  find  by  these  experiments  that  in  all  the  tubes  and  orifices  used  the 
ratio  of  the  velocity  at  the  smallest  section  to  the  velocity  due  the  head  is  least 
when  the  heads  are  very  small.  Thus  with  the  mouth-piece  A  alone, 

When  the  effective  head  is  0.0339  foot  (experiment  1),  the  ratio  is  0.8150 

«              "               «  0.2300    "  (         "  6),    «  «     0.8626 

«             "              "  0.9579    «  (         «  13),    «  «     0.9277 

"              "               "  1.5140  feet  (         «  17),    «  «     0.9407 


224  EXPERIMENTS   ON  THE   FLOW   OF   WATER 

With   the  mouth-piece  A  and  the  first  joint  B  of  the  diverging  tube, 

When  the  effective  head  is  0.0202  foot  (experiment  20),  the  ratio  is  1.1506 

«  «  «      0.0996     «  (          «  23),    "  "  1.3958 

«  «     0.8544    «  (         «          37),    "  "  1.5919 

«  "      1.4704  feet  (          «  38),     «  "  1.5748 

With  the  mouth-piece  .4  and  the  two  first  joints  B  and  'C  of  the  diverging 
tube, 

When  the  effective  head  is  0.0143  foot  (experiment  40),  the  ratio  is  1.4179 

«  «     0.0616    «  (          «          42),    "  «  1.7844 

«  1.0999  feet  (    «    49),  «     «  2.1643 

«      «       «  1.3116  "  (    "     50),  «"     "  2.1232 

With  the  mouth-piece  A  and  the  three  first  joints  B,  C,  and  D  of  the  diverg- 
ing tube, 

When  the  effective  head  is  0.0138  foot  (experiment  52),  the  ratio  is  1.3850 

«  «     0.0588    «  (          «  54),    «  «     1.8166 

«      «      «  1.1772  feet  (    «  62),  «     «  2.4306 

«      «      «  1.3612  «  (    «  63),  «     «  2.4266 

With   the   complete   compound   tube, 

When  the  effective  head  is  0.0139  foot  (experiment  67),  the  ratio  is  1.4381 

"  «  «     0.0575    «  (         «          70),    «  ,        «  1.8764 

«  «  «      1.2823  feet  (          «  78),    «  «  2.4213 

«  «  «     1.4085    «  (         «          79),    «  «  2.2672 

With  the  thin  plate, 

When  the  effective  head  is  0.0916  foot  (experiment  94),  the  ratio  is  0.5642 

"  «  "  0.4835    "  (         «          96),    «  «  0.5797 

«  «  «  1.0242  feet  (          «          99),    «  «  0.5915 

«  «  «  1.4903    «  (          «        101),    «  "  0.5922 

262.  By  the  preceding  extracts  from  table  XXVII.  it  will  be  seen  that  the 
ratio  of  the  velocity  at  the  smallest  section  of  the  tube  or  orifice  to  the  velocity 
due  the  head  is  the  least  when  the  effective  head  is  the  least,  and  in  the  cases  of 
the  mouth-piece  and  orifice  in  the  thin  plate,  the  ratio  is  the  greatest  when  the  effec- 
tive head  is  the  greatest. 


THKOUGH   SUBMERGED   ORIFICES   AND   DIVERGING   TUBES.  225 

In  the  case  of  the  diverging  tube,  the  value  of  the  ratio  is  a  maximum 
when  the  effective  head  is  somewhat  less  than  the  greatest. 

It  is  the  general  result  of  the  great  number  of  experiments  on  record,  on 
the  flow  of  water  through  orifices  in  a  thin  plate,  discharging  freely  into  the  air, 
that  the  coefficient  of  discharge  (which  in  simple  orifices  is  the  same  thing  as  the 
ratio  of  the  velocity  at  the  smallest  section  of  the  orifice  to  the  velocity  due  the 
head)  is  greatest  for  very  small  heads.  In  these  experiments  where  the  discharge 
takes  place  under  water,  the  coefficient  of  discharge  is  least  with  the  very  small 
heads.  This  result  is  so  marked  and  uniform  that  there  can  be  no  doubt  of  the 
fact. 

263.  As   to   the   value    of  the    coefficient   of  discharge    for   the   mouth-piece    A, 
a  mean   of    all    the    experiments   in   which   the    effective    head   is   not   less   than    1.5 
feet   gives    0.9451,   the   mean   effective   head   being  1.5150   feet.      This    is   nearly  the 
same    as   the  greatest  value   of  the    coefficient  of  discharge  found  by  Castel  for  the 
smallest   section  of  an  orifice  in  a  converging   conical   tube,  namely,   0.956,  which  is 
for   a   tube   in   which   the   sides   converge    at   an   angle    of    13°   40',  and    discharging 
freely  into  the  air.*     Michelotti,  in  one  of  his  experiments,  by  employing  a  cycloidal 
tube,  found  it  0.983.1      Eytelwein  found  0.9798.  t      Other  experimenters  have   found 
from    0.96    to    0.98.      We   must,    therefore,   conclude   that   the    coefficient  of  discharge 
for  the  mouth-piece  A,  when  discharging  under  water,  is  about  3  per  cent  less  than  has 
been  found  for   similar   orifices  when   discharging  freely  into  the  air. 

264.  The   value  of  the   coefficient   of  discharge   for   the  orifice  in  a  thin  plate, 
taking   the    mean    of    the    three    experiments    in   which   the    effective   head   is   near 
1.5   feet,   is    0.5920,   the   mean   effective    head   being    1.5009  feet.     This   is   less    than 
has   been   found   for   circular   orifices   in   a   thin  plate  discharging  freely  into  the  air. 
There    are    great    numbers    of    these    experiments   on   record,   made   with  orifices   of 
various   diameters   and    under   various   heads.     The    general   result   for  the    coefficient 
of  discharge  is  very  nearly  0.62.     We  must,  therefore,  conclude  that  the  flow  through 
a   submerged    orifice    in    a    thin   plate    is    less   than   when   the   discharge   takes   place 
freely  into   the  air,  in   the  ratio   of  0.59   to   0.62,   or  about   5  per  cent  less. 

265.  The   values   of  the    ratio    of  the    velocity   at   the    smallest   section    to   the 
velocity   due    the   head,   for   the    several    combinations    of    the   mouth-piece   and    the 
diverging    tube,    taking    the    largest    values    found    in    these     experiments,    are    as 
follows :  — 


*  D'Aubuisson's   Hydraulics,   Bennett's   translation,   page  56. 

f  MemoireS   de   1' Academic   Royale   des    Sciences  de  Turin,   1784-85. 

}  Handbuch   der   Mechanik    und   der   Hydraulik. 

29 


226  EXPERIMENTS   ON   THE   FLOW  OF   WATER 

For  the  mouth-piece  A  alone (exp.  91)  0.9507 

For  the  mouth-piece  A  and  the  first  joint  B  of  the  diverg- 
ing tube (   «     32)  1.5955 

For  the    mouth-piece    A    and    the    first    two  joints    B  and 

C  of  the  diverging  tube (    «     49)  2.1643 

For  the  mouth-piece  A  and  the  first  three  joints  B,  C,  and 

D  of  the  diverging  tube (   "     62)  2.4306 

For  the  complete  compound  tube    as  represented  by  figure 

2,   plate  XXI (   «     78)  2.4213 

The  maximum  effect  was  produced  with  the  mouth-piece  and  first  three  joints  of 
the  diverging  tube,  the  addition  of  the  fourth  joint  caused  a  slight  diminution.  In 
experiment  62,  giving  the  greatest  effect,  the  increase  in  the  velocity  of  the 
water  in  the  smallest  section  due  to  the  diverging  tubes  is  in  the  ratio  of  0.9507 
to  2.4306,  or  as  1  to  2.5566.  To  produce  this  increased  velocity  in  the  smallest 
section  without  using  the  diverging  tube  the  head  must  be  increased  in  the  ratio 
of  1  to  (2.5S66)2  or  as  1  to  6.5364.  The  effective  head  in  experiment  62  was 
1.1772  feet.  To  give  the  .velocity  in  the  same  experiment,  if  the  diverging  tube 
had  not  been  attached,  would  have  required  an  effective  head  of  1.1772  X  6.5364 
=  7.6947  feet.  The  difference  in  these  heads  is  7.6947  —  1.1772  =  6.5175  feet. 
A  portion  of  the  pressure  of  the  atmosphere  on  the  surface  of  the  water  in  the 
upper  division  E  of  the  cistern,  figures  1  and  2,  plate  XX.,  equivalent  to  this 
head  of  water,  is  rendered  active  by  the  addition  of  the  diverging  tube  to  the 
mouth-piece. 

266.     According   to    Bernoulli's   theory,   the    velocity   of    the    water   at   its   final 
discharge   from   the   tube    should   be    that   due  to  the  head;*    in  experiment  62  this 


*  Call  A  the  area  of  the  section  and  V  the  velocity  of  the  water  at  ab,  figure  2,  plate  XX.  B  the  area 
of  the  section  and  v  the  velocity  at  cd;  A  =  the  head  or  difference  of  height  of  the  surface  of  the  water  in 
compartments  E  and  F.  The  motion  having  become  permanent,  we  have 

A  V  =  Bv. 

The  volume  of  water  included  between  the  sections  ab  and  cd  in  the  small  time  t  will  move  .to  a' b'  c'  d' ; 
the  volume  included  between  the  sections  a'  b'  and  cd  is  common  to  both  positions,  every  particle  in  one 
having  its  counterpart  in  the  other,  both  in  position  and  velocity.  In  finding  the  change  in  the  living  force 
in  the  two  position?,  we  need  only  consider  the  volumes  a  a'  bb'  and  cc'  dd1.  These  volumes  are  equal, 
and  assuming  the  water  to  be  pure  and  at  its  maximum  density,  the  weight  of  each  is  62.382  A  Vt. 


THROUGH   SUBMERGED   ORIFICES   AND    DIVERGING   TUBES.  227 

velocity  is  8.7018  feet  per  second ;  the  velocity  at  other  parts  of  the  compound 
tube  would  be  inversely  as  the  squares  of  the  diameters ;  at  the  smallest  section 

the    velocity   must   be   greater   than    at    the    final    discharge    in    the    ratio    of    1    to 

/03209\2  * 

(  Q'  J  =  9.9367.  To  give  this  velocity  at  the  smallest  section  without  the  diverg- 
ing tube  would  require  the  effective  head  of  water  to  be  increased  from  1.1772 
feet  to  1.1772  X  (9.9367)2  =  116.24  feet;  the  increase  being  115.06  feet; 
if  the  pressure  of  the  atmosphere  was  great  enough,  its  pressure,  to  this 
extent,  would  be  rendered  active.  The  total  pressure  of  the  atmosphere  is  usually 
about  34  feet,  and  this  of  course  is  the  limit  to  which  it  can  be  rendered  active. 
Abstracting  from  the  effects  of  vaporization,  whenever  the  exhausting  effect  of  the 
diverging  tube  exceeds  the  pressure  of  the  atmosphere,  (added  to  the  pressure  due 
to  the  actual  head  of  water  at  the  smallest  section,)  breaks  must  occur  in  the 
mass  of  water  in  the  compound  tube,  at  or  near  the  smallest  section,  and  the 
flow  through  the  smallest  section  will  be  the  same  as  if  the  discharge  took  place 
in  a  vacuum.  In  experiment  62,  the  exhausting  effect  of  the  diverging  tube, 


„,....,  .  ,,  .   ,  ,.  .      62.382^1  Vt 

The  living  force  of  the  volume  a  a'  bb1  is   - 

cc,dd,  is       - 


9 
The  increase  of  living  force  in  passing  from  one  position  to  the  other  being 

68.882^  Fl()),_^  (1) 

if 

This  increase  of  living  force  is  produced  by  the  action  of  gravity  on  the  volume  of  water  A  Vt  descending 
through  the  height  h,  which  is  equivalent  to  an  amount  of  work  represented  by 

62.382^1  Vth.  (2.) 

• 
By  the  doctrine  of  living  forces,  the  living  force  (1.)  is  equivalent  to  the  amount  of  work  represented  by 

62.382  A  Vt(v*_v^  (3-) 

The  amount  of  work  in  (2.)  and  (3.)  must  be  equal  ;  we  have,  therefore, 

62.382  AVth  =  ™ 


from  which  we  deduce  h  = 


-  - 

If  V  is  very  small  relatively  to  v,  it  may  be  neglected,  and  we  have 

v' 


h  =  —~  ,  and  v  =  \/2gh. 
y 


228  EXPERIMENTS   ON   THE   FLOW   OF    WATER 

according  to  Bernoulli's  theory,  exceeds  three  times  the  actual  pressure  at  the 
smallest  section,  and  if  it  had  produced  its  full  effect  according  to  theory  or  even 
one  third  of  that  effect,  breaks  must  have  occurred  in  the  mass  of  water  near  the 
smallest  section. 

The  ratio  of  the  actual  velocity  of  the  water  at  its  final  discharge  to  the 
velocity  according  to  Bernoulli's  theory  is  given  in  column  17.  In  experiment 
62  it  is  0.2446,  or  about  one  quarter  of  the  velocity  due  the  head,  indicating  a 
loss  of  about  fifteen  sixteenths  of  the  living  force.  It  is  difficult  to  see  how  so 
much  can  be  lost.  There  are  no  abrupt  changes  in  velocity,  and  the  interior 
surfaces  of  the  mouth-piece  and  diverging  tube  are  smooth  and  free  from  sensible 
irregularity.  The  slight  oxidation  observable  after  some  of  the  experiments  appears 
to  have  produced  no  sensible  loss,  as  in  experiment  62,  which  gave  the  greatest 
result,  there  was  considerable  oxidation,  while  in  other  experiments  giving  a  less 
effect  there  was  no  oxidation. 

The  chief  discrepancy  between  the  hypothesis  on  which  Bernoulli's  theory  is 
founded  and  the  real  conditions  of  the  motion  appears  to  be  due  to  the  retard- 
ing effects  of  the  walls  of  the  tube.  According  to  the  hypothesis,  the  velocity  in 
all  parts  of  the  same  section  is  the  same ;  Prony's  well-known  formula  for  the 
motion  of  water  in  pipes  is  founded  upon  the  idea  that  the  principal  retardation 
is  due  to  the  sides ;  whence  it  follows,  that  the  velocity  must  be  least  at  the 
sides  and  greatest  at  the  centre.  Darcy*  made  many  experiments  on  the  subject 
by  means  of  Pitot's  tube,  and  found  that  in  long  straight  pipes  there  was  a 
material  variation  in  the  velocities  at  different  distances  from  the  centre,  and 
determined  a  formula  expressing  the  law  of  the  variation.  It  would  not  be  safe 
to  apply  this  formula  to  these  experiments  on  account  of  the  short  length  and 
varying  diameter  of  the  compound  tube,  but  it  is  clear  that  variations  in  the 
velocity  must  exist  to  an  extent  which  must  greatly  modify  the  results  deduced 
from  Bernoulli's  theory. 

267.  As  previously  stated,  Venturi,  by  adding  a  diverging  tube  increased  the  dis- 
charge of  an  orifice  having  nearly  the  form  of  the  contracted  vein,  and  discharging 
freely  into  the  air,  in  the  ratio  of  1  to  2.21.  In  these  experiments,  in  an  orifice 
without,  contraction  discharging  under  water  the  discharge  was  increased  by  adding 
a  diverging  tube  in  the  ratio  of  1  to  2.56.  Making  the  comparison  with  an  orifice  in  a 
thin  plate,  the  maximum  coefficient  of  discharge  with  the  thin  plate  is  0.5928,  and 
with  the  month-piece  of  cycloidal  form  and  diverging  tube,  the  maximum  coefficient 


*  Recherches   experimentales   relatives  au   Mouvement   de   tEau   dans   las    Tuyaux,   par   HENET   DAKCT. 
Paris,  1857, 


THROUGH   SUBMERGED   ORIFICES   AND   DIVERGING   TUBES. 


229 


is   2.4306 ;   the    discharge  with   the    same    area   of  orifice    and   the    same   head   being 
increased  in  the  ratio  of  1  to  4.12. 

268.  Considerable  irregularities  will  be  observed  in  the  value  of  the  ratio  of 
the  velocity  in  the  smallest  section  to  the  velocity  due  the  head,  given  in  column 
14.  Thus,  in  the  experiments  with  the  complete  compound  tube,  we  have  the 
following,  which  were  intended  to  be  identical,  the  repetitions  being  made  for  the 
purpose  of  detecting  such  variations,  if  any  should  occur.  "In  all  these  experiments 
the  index  of  the  inlet  cock,  L,  figures  2  and  3,  plate  XX.,  was  set  at  the  same 
point,  viz.  62.5°,  or  as  nearly  so  as  practicable,  in  order  to  admit  the  same  quan- 
tity of  water. 


Number  of  the 
experiment  ill 
Table  XXVII. 

Quantity  of  water 
discharged  ;  in 
Cubic  feet  per  second. 

Effective  head  pro- 
ducing the  discharge  ; 
iu  feet. 

Ratio  of  the  Telocity 
at  the  smallest  section 
to  the  velocity  due 
the  head. 

65 

0.17574 

1.4183 

2.2606 

66 

0.17390 

1.4073 

2.2456 

78 

0.17898 

1.2823 

2.4213 

79 

0.17565 

1.4085 

2.2672 

88 

0.17630 

1.3769 

2.3016 

89 

0.17713 

1.3612 

2.3257 

90 

0.17736 

1.3449 

23428 

92 

0.17713 

1.2863 

2.3925 

[n  the  preceding  table,  the  small  irregularities  in  the  quantities  of  water  dis- 
charged are  due  to  corresponding  small  variations  in  setting  the  index  of  the  inlet 
cock.  The  irregularities  in  the  effective  head  are  mainly  due  to  changes  in  the 
efficiency  of  the  diverging  tube.  The  only  known  variation  on  which  these  changes 
could  depend  is  in  the  state  of  the  interior  surface  of  the  tube.  Thus  No.  65 
was  the  second  experiment  made  after  the  grease  was  wiped  off.  Twelve  exper- 
iments were, made  between  Nos.  65  and  78,  no  change  being  made  in  the  state 
of  the  surface,  except  that  caused  by  the  action  of  the  water,  which  undoubtedly 
had  washed  off,  before  No.  78  was  made,  a  part  or  the  whole  of  the  grease  not 
removed  by  wiping.  In  the  experiments  made  soon  after  wiping  the  surface,  it  is 
probable  that  the  water  was  repelled  from  it  by  the  grease,  but  after  the  water 
had  run  through  the  tube  for  some  hours  the  grease  was  washed  off  sufficiently 
to  permit  the  water  to  come  in  contact  with  the  iron,  which  appears  to  have 
increased,  materially,  the  exhausting  effect  of  the  diverging  tube. 

269.  Previous  to  making  the  experiments,  it  was  anticipated  that  when  the 
diverging  tube  was  used  there  would  be  sensible  oscillations  in  the  elevation  of 
the  surface  of  the  water  in  compartment  E,  figures  1  and  2,  plate  XX.,  due  to 
the  unstable  equilibrium  of  the  stream.  Although  the  amplitudes  of  the  oscillations 


230  EXPERIMENTS   ON  THE   FLOW  OF   WATER 

of  the  surface  were  much  less  than  was  expected,  they  were  quite  sensible.  Thus 
we  find,  by  referring  to  the  original  notes,  that  with  the  mouth-piece  alone,  the 
amplitude  of  the  oscillations, 

when  the  effective  head  was  0.10  foot,  was  about  0.0003  foot 
"       "  «  "        "     1.00     "        "         "      0.0006     " 

«  "  "        "     1.40  feet      "        "      0.0007     " 

With   the    complete    compound    tube    the    amplitude    of  the    oscillations, 

when  the  effective  head  was  0.10  foot,  was  about  0.0021  foot. 

"  "        "     1.00     "        "        "      0.0103     " 

«  "       "     1.40  feet     "        "      0.0117    " 

The  variation  with  heads  from  1.00  foot  to  1.40  feet  being  about  17  times  as  great 
with  the  complete  diverging  tube  as  with  the  mouth-piece  alone. 


270.  As  previously  stated,  the  principles  involved  in  the  flow  of  water  through 
a  diverging  tube  find  a  useful  application  in  Mr.  Boyden's  Diffuser.  This  inven- 
tion, applied  to  a  turbine  water- wheel  104.25  inches  in  diameter  and  about  seven 
hundred  horse  power,  is  represented  in  plates  XXII.  and  XXIII.  This  turbine  is 
one  of  four  of  the  same  power  constructed  from  the  designs  of  the  author  for 
the  cotton-mills  of  the  Merrimack  Manufacturing  Company  in  Lowell.  Plate  XXII. 
is  a  sectional  elevation  through  the  axis,  showing  the  lower  parts  of  the  apparatus, 
a,  a,  a,  a  is  the  wheel,  carrying  60  floats  of  Russian  sheet  iron,  0.15  inch  thick;  6 
the  main  shaft,  which  is  suspended  from  the  top,  in  a  similar  manner  to  the  Tremont 
turbines  (plate  I.);  c,  c  is  the  disc,  carrying  33  guides,  c,  c,  c',  c,  of  Russian  sheet  iron, 
0.125  inch  thick,  which  lean  one  horizontally  to  six  vertically ;  d,  d,  the  disc  pipe,  which 
hangs  at  its  upper  end,  upon  a  part  of  the  curved  pipe  or  curb  e,  e,  not  represented  in  the 
plate ;  /,  /,  the  garniture,  which  supports  the  upper  part  of  the  guides,  and  is 
curved  at  its  lower  edge,  in  order  to  afford  a  favorable  aperture  for  the  flow  of 
the  water  entering  the  wheel ;  g,  g,  the  lower  curb ;  h,  h,  the  speed  gate,  which  is 
represented  as  raised  to  its  greatest  height;  i,  a  gate  rod,  which  with  two  others, 
not  represented  in  the  plate,  enables  the  gate  to  be  moved  by  the  governor  or 
by  hand ;  k,  k,  beams  extending  from  the  granite  walls  of  the  wheel-pit  to  the 
lower  curb  and  supporting  the  latter;  I,  I,  pillars  resting  upon  granite  blocks  in 
the  floor  of  the  wheel-pit,  and  supporting  the  beams  k,  k;  m,  m,  the  diffuser,  which 
is  supported  by  the  pillars  I,  I,  by  means  of  the  curved  beams  n,  n,  n,  n ;  w,  w,  low 
water  level  of  the  surface  of  the  water  in  the  wheel-pit.  The  wheel  is  placed  suf- 


THROUGH   SUBMERGED   ORIFICES   AND   DIVERGING   TUBES.  231 

ficiently  low,  to  permit  the  diffuser  to  be  submerged  at  all  times  when  the  wheel  is 
in  operation,  that  being  essential  to  the  most  advantageous  operation  of  the  diffuser. 
Figure  1,  plate  XXIII.,  is  a  horizontal  section  through  the  wheel,  showing  also 
the  disc,  guides,  and  garniture,  and  also  the  lower  part  of  the  diffuser.  Figure 
2  is  a  horizontal  section  on  a  larger  scale,  showing  part  of  the  wheel,  guides, 
and  diffuser.  Figure  3  is  a  vertical  section,  showing  part  of  the  wheel,  diffuser,  &c. 
When  the  speed  gate  is  fully  raised,  and  the  wheel  is  moving  with  the 
velocity  giving  its  greatest  coefficient  of  useful  effect,  the  water  passes  through  the 
wheel  in  a  path,  which  is  nearly  represented  by "  the  dotted  line  a,  b,  figure  2, 
plate  XXIII.  On  leaving  the  wheel  it  necessarily  has  considerable  velocity,  which 
would  involve  a  corresponding  loss  of  power,  except  for  the  effect  of  the  diffuser, 
which  utilizes  a  portion  of  it.  When  operating  under  a  fall  of  33  feet  and  the 
speed  gate  raised  to  its  full  height,  this  wheel  discharges  about  219  cubic  feet  of 
water  per  second.  The  area  of  the  annular  space  o,  o,  o,  o,  plate  XXII.,  where 

the   water  enters   the   diffuser,  is   0.802  X  8.792  TT  =  22.152  square  feet;    and   if  the 

219 
stream  passes  through  this  section  radially,  its  mean  velocity  must  be  22  52  =  9.886 

feet  per  second,  which  is  due  to  a  head  of  1.519  feet.  The  area  of  the  annular 
space  p,  p,  p,  p,  where  the  water  leaves  the  diffuser,  is  1.5  X  15.333  TT  =  72.255 

219 
square   feet,  and    the   mean  velocity  =  3.031  feet  per  second,  which  is  due  to 

a  head  of  0.143  feet.  According  to  this,  the  saving  of  head,  due  to  the  diffuser  is 
1.519  _  0.143  =  1.376  feet,  being  9,  L3/S,o>  or  about  4I  Per  cent  of  tne  head 

OO. 1 .0  1  J 

available  without  the  diffuser,  which  is  equivalent  to  a  gain  in  the  coefficient  of 
useful  effect  to  the  same  extent.  As  previously  stated  (art.  12),  experiments  on  the 
same  turbine,  with  and  without  a  diffuser,  have  shown  a  gain  due  to  the  latter,  of 
about  3  per  cent  in  the  coefficient  of  useful  effect.  The  diffuser  adds  to  the  co- 
efficient of  useful  effect  by  increasing  the  velocity  of  the  water  passing  through 
the  wheel,  and  it  must  of  course  increase  the  quantity  of  water  discharged  in 
the  same  proportion.  If  it  increases  the  available  head  3  per  cent,  the  velocity, 
which  varies  as  the  square  root  of  the  head,  must  be  increased  about  1.5  per 
cent,  and  the  quantity  discharged  must  be  increased  in  the  same  proportion. 
The  power  of  the  wheel,  which  varies  as  the  product  of  the  head  into  the  quantity  of 
water  discharged,  must  be  increased  about  4.5  per  cent. 


232 


EXPLANATION   OF  TABLES  XXVIII.,   XXIX.,  AND  XXX. 

These  tables  have  been  prepared  in  the  office  of  the  Proprietors  of  the  Locks 
and  Canals  on  Merrimack  River,  for  the  purpose  of  facilitating  the  computations 
connected  with  gauging  the  quantities  of  water  drawn  from  their  canals  at 
Lowell. 

TABLE  XXVIII.  gives  the  velocities  of  floats  for  eight  different  distances  be- 
tween the  transit  stations,  and  for  times  of  passage  between  them  for  every  tenth 
of  a  second,  from  20  to  100  seconds. 

The  use  of  the  table  may  be  extended  to  such  other  distances  between  the 
transit  stations  as  are  multiplies  or  submultiplies  of  the  distances  given  in  the  table, 
by  taking  the  time  the  same  multiple  or  submultiple  as  the  distance. 

TABLE  XXIX.  gives  the  values  of  the  coefficient  (l  —  0.116  (\I~T)  —  0.1))  for 
values  of  D  for  every  0.001  from  0.000  to  0.100,  with  the  logarithms  of  the  same. 
(See  art.  233.) 

TABLE  XXX.  gives  the  velocities,  in  feet  per  second,  due  to  every  0.01  foot 
head,  from  0.00  to  49.99  feet,  computed  for  Lowell,  by  the  formulas  given  in 
art.  68.  These  formulas,  reduced  to  the  English  foot  as  the  unit,  become 

g  =  32.1695  (1  -  -  0.00284  cos.  2  1)  (i  - 
r  =  20887540  (1  +  0.00164  cos.  2  I). 

The  values  of  g  by  these  formulas  for  several  latitudes  and  heights  above 
the  sea.  are  given  in  the  following  table :  — 


Height 
above  the 
Sea. 

Feet. 

Latitude. 

300 

350 

400 

45° 

5O° 

55° 

600 

0 
100 
200 
300 

32.1239 
32.1236 
32.1233 
32.1229 

32.1383 
32.1380 
32.1377 
32.1374 

32.1537 
32.1534 
32.1531 
32.1528 

32.1695 
32.1692 
32.1689 
32.1686 

32.1854 
32.1851 
32.1848 
32.1845 

32.2008 
,  32.2005 
32.2002 
32.1998 

32.2152 
32.2149 
32.2146 
32.2143 

400 
500 
600 
700 

32.1226 
32.1223 
32.1220 
32.1217 

32.1371 
32.1368 
32.1364 
32.1361 

32.1524 
32.1521 
32.1518 
32.1515 

32.1683 
32.1680 
32.1677 
32.1674 

32.1842 
32.1839 
32.1835 
32.1832 

32.1995 
32.1992 
32.1989 
32.1986 

32.2140 
32.2137 
32.2134 
32.2131 

800 
900 
„  1000 
1100 

32.1214 
32.1211 
32.1208 
32.1205 

32.1358 
32.1355 
32.1352 
32.1349 

32.1512 
32.1509 
32.1506 
32.1503 

32.1671 
32.1668 
32.1665 
32.1662 

32.1829 
32.1826 
32.1823 
32.1820 

32.1983 
32.1980 
32.1977 
32.1974 

32.2128 
32.2125 
32.2121 
32.2118 

233 
TABLE    XXVIII. 

TABLE    OF    VELOCITIES    OP    TUBES   IN  MEASURING    FLUMES,    IN    FEET    PER    SECOND.       THE    TIME 

OCCUPIED   IN  PASSING   FROM   THE   UPSTREAM   TO   THE    DOWNSTREAM   TRANSIT 

STATION,  AND  THE   DISTANCE  BETWEEN   THEM,  BEING  GIVEN. 


TIME. 

DISTANCE  BETWEEN  THE  TRANSIT  STATIONS.  IN  FEET. 

TIME 

DISTANCE   BETWEEN  THE  TRANSIT  STATIONS,  IN  FEET. 

See's. 

50. 

1  60. 

70. 

80. 

90. 

100. 

110. 

120. 

See's. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

20.0 

2.500 

3.000 

3.500 

4.000 

4.500 

5.000 

5.500 

6.000 

25.0 

2.000 

2.400 

2.800 

3.200 

3.600]  4.000 

4.400 

4.800 

20.1 

2.488 

2.985 

3.483 

3.980  4.478 

4.975 

5.473 

5.970 

25.1 

1.992 

2.390 

2.789 

3.187 

3.586|  3.984 

4.382 

4.781 

20.2 

2.475 

2.970 

3.465 

3.960 

4.455 

4.950 

5.446 

5.941 

25.2 

1.984 

2.381 

2.778 

3.175 

3.571 

3.968 

4.365 

4.762 

20.3 

2.463 

2.956 

3.448 

3.941 

4.433 

4.926 

5.419 

5.911 

25.3 

1.976 

2.372 

2.767 

3.162 

3.557 

3.953 

4.348 

4.743 

20.4 

2.451 

2.941 

3.431 

3.922 

4.412 

4.902 

5.392 

5.882 

25.4 

1.969 

2.362 

2.756 

3.150 

3.543 

3.937 

4.331 

4.724 

20.5 

2.439 

2.927 

3.415 

3.902 

4.390 

4.878 

5.366 

5.854 

25.5 

1.961 

2.353 

2.745 

3.137 

3.529 

3.922 

4.314 

4.706 

20.6 

2.427 

2.913 

3.398 

3.883 

4.369 

4.854 

5.340 

5.825 

25.6 

1.953 

2.344 

2.734 

3.125 

3.516 

3.906 

4.297 

4.687 

20.7 

2.415 

2.899 

3.382 

3.865 

4.348 

4.831 

5.314 

5.797 

25.7 

1.946 

2.335 

2.724 

3.113 

3.502 

3.891 

4.280 

4.669 

20.8 

2.404 

2.885 

3.365 

3.846 

4.327 

4.808 

5.288 

5.769 

25.8 

1.938 

2.326 

2.713 

3.101 

3.488 

3.876 

4.264 

4.651 

20.9 

2.392 

2.871 

3.349 

3.828 

4.300 

4.785 

5.263 

5.742 

25.9 

1.931 

2.317 

2.703 

3.089 

3.475 

3.861 

4.247 

4.633 

21.0 

2.381 

2.857 

3.333 

3.810 

4.286 

4.762 

5.238 

5.714 

26.0 

1.923 

2.308 

2.692 

3.077 

3.462 

3.846 

4.231 

4.615 

21.1 

2.370 

2.844 

3.318 

3.791 

4.265 

4.739 

5.213 

5.687 

26.1 

1.916  2.299 

2.682  3.065 

3.448 

3.831 

4.215 

4.598 

21.2 

2.358 

2.830 

3.302 

3.774 

4.245 

4.717 

5.189 

5.660 

26.2 

1.908 

2.290 

2.672 

3.053 

3.435 

3.817 

4.198 

4.580 

21.3 

2.347 

2.817 

3.286 

3.756 

4.225 

4.695 

5.164 

5.634 

26.3 

1.901 

2.281 

2.662 

3.042 

3.422 

3.802 

4.183 

4.563 

21.4 

2.336 

2.804 

3.271 

3.738 

4.206 

4.673 

5.140 

5.607 

26.4 

1.894 

2.273 

2.652 

3.030 

3.409 

3.788 

4.167 

4.545 

21.5 

2.326 

2.791 

3.256 

3.721 

4.186 

4,651 

5.116 

5.581 

26.5 

1.887 

2.264 

2.642 

3.019 

3.396 

3.774 

4.151 

4.528 

21.6 

2.315 

2.778 

3.241 

3.704 

4.167 

4.630 

5.093 

5.556 

26.6 

1.880 

2.256 

2.632 

3.008 

3.383 

3.759 

4.135 

4.511 

21.7 

2.304 

2.765 

3.226 

3.687 

4.147 

4.608 

5.069 

5.530 

26.7 

1.873 

2.247 

2.622 

2.996 

3.371 

3.745 

4.120 

4.494 

21.8 

2.294 

2.752 

3.211 

3.670 

4.128 

4.587 

5.046 

5.505 

26.8 

1.866 

2.239 

2.612 

2.985 

3.358 

3.731 

4.104 

4.478 

21.9 

2.283 

2.740 

3.196 

3.653 

4.110 

4.566 

5.023 

5.479 

26.9 

1.859 

2.230 

2.602 

2.974 

3.340 

3.717 

4.089 

4.461 

22.0 

2.273 

2.727 

3.182 

3.636 

4.091 

4.545 

5.000 

5.455 

27.0 

1.852 

2.222 

2.593 

2.963 

3.333 

3.704 

4.074 

4.444 

22.1 

2.262 

2.715 

3.167 

3.620 

4.072 

4.525 

4.977 

5.430 

27.1 

1.845 

2.214 

2.583 

2.952 

3.321 

3.690 

4.059 

4.428 

22.2 

2.252 

2.703 

3.153 

3.604 

4.054 

4.505 

4.955 

5.405 

27.2 

1.838 

2.206 

2.574 

2.941 

3.309 

3.676 

4.044 

4.412 

22.3 

2.242 

2.691 

3.139 

3.587 

4.036 

4.484 

4.933 

5.381 

27.3 

1.832 

2.198 

2.564 

2.930 

3.297 

3.663 

4.029 

4.396 

22.4 

2.232 

2.679 

3.125 

3.571 

4.018 

4.464 

4.911 

5.357 

27.4 

1.825 

2.190 

2.555 

2.920 

3.285 

3.650 

4.015 

4,380 

22.5 

2.222 

2.667 

3.111 

3.556 

4.000 

4.444 

4.889 

5.333 

27.5 

1.818 

2.182 

2.545 

2.909 

3.273 

3.636 

4.000 

4.364 

22.6 

2.212 

2.655 

3.097 

3.540 

3.982 

4.425 

4.867 

5.310 

27.6 

1.812 

2.174 

2.536 

2.899 

3.261 

3.623 

3.986 

4.348 

22.7 

2.203 

2.643 

3.084 

3.524 

3.965 

4.405 

4.846 

5.286 

27.7 

1.805 

2.166 

2.527 

2.888 

3.249 

3.610 

3.971 

4.332 

22.8 

2.193 

2.632 

3.070 

3.509 

3.947 

4.386 

4.825 

5.263 

27.8 

1.799 

2.158 

2.518 

2.878 

3.237 

3.597 

3.957 

4.317 

22.9 

2.183 

2.620 

3.057 

3.493 

3.930 

4.367 

4.803 

5.240 

27.9 

1.792 

2.151 

2.509 

2.867 

3.226 

3.584 

3.943 

4.301 

23.0 

2.174 

2.C09 

3.043 

3.478 

3.913 

4.348 

4.783 

5.217 

28.0 

1.786 

2.143 

2.500 

2.857 

3.214 

3.571 

3.929 

4.28C. 

23.1 

2.165 

2.597 

3.030 

3.463 

3.896 

4.329 

4.762 

5.195 

28.1 

1.779 

2.135 

2.491 

2.847 

3.203 

3.559 

3.915 

4.270 

23.2 

2.155 

2.586 

3.017 

3.448 

3.879 

4.310 

4.741 

5.172 

28.2 

1.773 

2.128 

2.482 

2.837 

3.191 

3.546 

3.901 

4.255 

23.3 

2.146 

2.575 

3.004 

3.433 

3.863 

4.292 

4.721 

5.150 

28.3 

1.767 

2.120 

2.473 

2.827 

3.180 

3.534 

3.887 

4.240 

23.4 

2.137 

2.564 

2.991 

3.419 

3.846 

4.274 

4.701 

5.128 

28.4 

1.761 

2.113 

2.465 

2.817 

3.169 

3.521 

3.873 

4.225 

23.5 

2.128 

2.553 

2.979 

3.404 

3.830 

4.255 

4.681 

5.106 

28.5 

1.754 

2.105 

2.456 

2.807 

3.158 

3.509 

3.860 

4.211 

23.6 

2.119 

2.542 

2.966 

3.390 

3.814 

4.237 

4.661 

5.085 

28.6 

1.748 

2.098 

2.448 

2.797 

3.147 

3.497 

3.846 

4.196 

23.7 

2.110 

2.532 

2.954 

3.376 

3.797 

4.219 

4.641 

5.063 

28.7 

1.742 

2.091 

2.439 

2.787 

3.136 

3.484 

3.833 

4.181 

23.8 

2.10112.521 

2.941 

3.361 

3.782 

4.202 

4.622 

5.042 

28.8 

1.736 

2.083 

2.431 

2.778 

3.125 

3.472 

3.819 

4.167 

23.9 

2.092 

2.510 

2.929 

3.347 

3.766 

4.184 

4.603 

5.021 

28.9 

1.730 

2.076 

2.422 

2.768 

3.114 

3.460 

3.806 

4.152 

24.0 

2.083 

2.500 

2.917 

3.333 

3.750 

4.167 

4.583 

5.000 

29.0 

1.724 

2.069 

2.414 

2.759 

3.103 

3.448 

3.793 

4.138 

24.1 

2.075 

2.490 

2.905 

3.320 

3.734 

4.149 

4.564 

4.979 

29.1 

1.718 

2.062 

2.405 

2.749 

3.093 

3.436 

3.780 

4.124 

24.2 

2.066 

2.479 

2.893 

3.306 

3.719 

4.132 

4.545 

4.959 

29.2 

1.712 

2.055 

2.397 

2.740 

3.082 

3.425 

3.767 

4.110 

24.3 

2.058 

2.469 

2.881 

3.292 

3.704 

4.115 

4.527 

4.938 

29.3 

1.706 

2.048 

2.389 

2.730 

3.072 

3.413 

3.754 

4.096 

24.4 

2.049 

2.459 

2.869 

3.279 

3.689 

4.098 

4.508 

4.918 

29.4 

1.701 

2.041 

2.381 

2.721 

3.061 

3.401 

3.741 

4.082 

24.5 

2.041 

2.449 

2.857 

3.265 

3.673 

4.082 

4.490 

4.898 

29.5 

1.695 

2.034 

2.373 

2.712 

3.051 

3.390 

3.729 

4.068 

24.6 

2.033 

2.439 

2.846 

3.252 

3.659 

4.065 

4.472 

4.878 

29.6 

1.689 

2.027 

2.365 

2.703 

3.041 

3.378 

3.716 

4.054 

24.7 

2.024 

2.429 

2.834 

3.239 

3.644 

4.049 

4.453 

4.858 

29.7 

1.684 

2.020 

2.357 

2.694 

3.030 

3.367 

3.704 

4.040 

24.8 

2.016 

2.419 

2.823 

3.226 

3.629 

4.032 

4.435 

4.839 

29.8 

1.678 

2.013 

2.349 

2.685 

3.020 

3.356 

3.691 

4.027 

24.9 

2.008 

2.410 

2.811 

3.213 

3.614 

4.016 

4.418 

4.819 

29.9 

1.672  2.007 

2.341 

2.676 

3.010 

3.344 

3.679 

4.013 

234 
TABLE    XXVIII  — CONTINUED. 

TABLE    OF    VELOCITIES    OF    TUBES    IN  MEASURING    FLUMES,    IN    FEET   PER    SECOND.       THE    TIME 

OCCUPIED  IN   PASSING   FROM   THE   UPSTREAM   TO   THE    DOWNSTREAM   TRANSIT 

STATION,  AND  THE   DISTANCE  BETWEEN   THEM,  BEING  GIVEN. 


TIME. 

See's. 

DISTANCE  BETWEEN  THE  TRANSIT  STATIONS,  IN  FEET. 

TIME. 

See's. 

DISTANCE  BETWEEN  TUB  TRANSIT  STATIONS,  IN  FEET. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

30.0 

1.667 

2.000 

2.333 

2.667 

3.000 

3.333 

3.667 

4.000 

35.0 

1.429 

1.714 

2.000 

2.286 

2.571 

2.857 

3.143 

3.429 

30.1 

1.661 

1.993 

2.326 

2.658 

2.990 

3.322 

3.654 

3.987 

35.1 

1.425 

1.709 

1.994 

2.279 

2.564 

2.849 

3.134 

3.419 

30.2 

1.656 

1.987 

2.318 

2.649 

2.980 

3.311 

3.642 

3.974 

35.2 

1.420 

1.705 

1.989 

2.273 

2.557 

2.841 

3.125 

3.409 

30.3 

1.650 

1.980 

2.310 

2.640 

2.970 

3.300 

3.630 

3.960 

35.3 

1.416 

1.700 

1.983 

2.266 

2.550 

2.833 

3.116 

3.399 

30.4 

1.645 

1.974 

2.303 

2.632 

2.961 

3.289 

3.618 

3.947 

35.4 

1.412 

1.695 

1.977 

2.260 

2.542 

2.825 

3.107 

3.390 

30.5 

1.639 

1.967 

2.295 

2.623 

2.951 

3.279 

3.607 

3.934 

35.5 

1.408 

1.690 

1.972 

2.254 

2.535 

2.817 

3.099 

3.380 

30.6 

1.634 

1.961 

2.288 

2.614 

2.941 

3.268 

3.595 

3.922 

35.6 

1.404 

1.685 

1.966 

2.247 

2.528 

2.809 

3.090 

3.371 

30.7 

1.629 

1.954 

2.280 

2.606 

2.932 

3.257 

3.583 

3.909 

35.7 

1.401 

1.681 

1.961 

2.241 

2.521 

2.801 

3.081 

3.361 

30.8 

1.623 

1.948 

2.273 

2.597 

2.922 

3.247 

3.571 

3.896 

35.8 

1.397 

1.676 

1.955 

2.235 

2514 

2.793 

3.073 

3.352 

30.9 

1.618 

1.942 

2.265 

2.589 

2.913 

3.236 

3.560 

3.883 

35.9 

1.393 

1.671 

1.950 

2.228 

2.507 

2.786 

3.064 

3.343 

31.0 

1.613 

.935 

2.258 

2.581 

2.903 

3.226 

3.548 

3.871 

36.0 

1.389 

1.667 

1.944 

2.222 

2.500 

2.778 

3.056 

3.333 

31.1 

1.608 

.929 

2.251 

2.572 

2.894 

3.215 

3.537 

3.859 

36.1 

1.385 

1.662 

1.939 

2.216 

2.493 

2.770 

3.047 

3.324 

31.2 

1.603 

.923 

2.244 

2.564 

2.885 

3.205 

3.526 

3.846 

36.2 

1.381 

1.657 

1.934 

2.210 

2.486 

2.762 

3.039 

3.315 

31.3 

1.597 

.917 

2.236 

2.556 

2.875 

3.195 

3.514 

3.834 

36.3 

1.377 

1.653 

1.928 

2.204 

2.479 

2.755 

3.030 

3.306 

31.4 

1.592 

.911 

2.229 

2.548 

2.866 

3.185 

3.503 

3.822 

36.4 

1.374 

1.648 

1.923 

2.198 

2.473 

2.747 

3.022 

3.29V7 

31.5 

1.587 

.905 

2.222 

2.540 

2.857 

3.175 

3.492 

3.810 

36.5 

1.370 

1.644 

1.918 

2.192 

2.466 

2.740 

3.014 

3.288 

31.6 

1.582 

.899 

2.215 

2.532 

2.848 

3.165 

3.481 

3.797 

36.6 

1.366 

1.639 

1.913 

2.186 

2.459 

2.732 

3.005 

3.279 

31.7 

1.577 

1.893 

2.208 

2.524 

2.839 

3.155 

3.470 

3.785 

36.7 

1.362 

1.635 

1.907 

2.180 

2.452 

2.725 

2.997 

3.270 

31.8 

1.572 

1.887 

2.201 

2.516 

2.830 

3.145 

3.459 

3.774 

36.8 

1.359 

1.630 

1.902 

2.174 

2.446 

2.717 

2.989 

3.261 

31.9 

1.567 

1.881 

2.194 

2.508 

2.821 

3.135 

3.448 

3.762 

36.9 

1.355 

1.626 

1.897 

2.168 

2.439 

2.710 

2.981 

3.252 

32.0 

1.562 

1.875 

2.187 

2.500 

2.812 

3.125 

3.437 

3.750 

37.0 

1.351 

1.622 

1.892 

2.162 

2.432 

2.703 

2.973 

3.243 

32.1 

1.558 

1.869 

2.181 

2.492 

2.804 

3.115 

3.427 

3.738 

37.1 

1.348 

1.617 

1.887 

2.156 

2.426 

2.695 

2.965 

3.235 

32.2 

1.553 

1.803 

2.174 

2.484 

2.795 

3.106 

3.416 

3.727 

37.2 

1.344 

1.613 

1.882 

2.151 

2.419 

2.688 

2.957 

3.226 

32.3 

1.548 

1.858 

2.167 

2.477 

2.786 

3.096 

3.406 

3.715 

37.3 

1.340 

1.609 

1.877 

2.145 

2.413 

2.681 

2.949 

3.217 

32.4 

1.543 

1.852 

2.160 

2.469 

2.778 

3.086 

3.395 

3.704 

37.4 

1.337 

1.604 

1.872 

2.139 

2.406 

2.674 

2.941 

3.209 

32.5 

1.538 

1.846 

2.154 

2.462 

2.769 

3.077 

3.385 

3.692 

37.5 

1.333 

1.600 

1.867 

2.133 

2.400 

2.667 

2.933 

3.200 

32.6 

1.534 

1.840 

2.147 

2.454 

2.761 

3.067 

3.374 

3.681 

37.6 

1.330 

1.596 

1.862 

2.128 

2.394 

2.660 

2.926 

3.191 

32.7 

1.529 

1.835 

2.141 

2.446 

2.752 

3.058 

3.364 

3.670 

37.7 

1.326 

1.592 

1.857 

2.122 

2.387 

2.653 

2.918 

3.183 

32.8 

1.524 

1.829 

2.134 

2.439 

2.744 

3.049 

3.354 

3.659 

37.8 

1.323 

1.587 

1.852 

2.116 

2.381 

2.646 

2.910 

3.175 

32.9 

1.520 

1.824 

2.128 

2.432 

2.736 

3.040 

3.343 

3.647 

37.9 

1.319 

1.583 

1.847 

2.111 

2.375 

2.639 

2.902 

3.166 

33.0 

1.515 

1.818 

2.121 

2.424 

2.727 

3.030 

3.333 

3.636 

38.0 

1.316 

1.579 

1.842 

2.105 

2.368 

2.632 

2.895 

3.158 

33.1 

1.511 

1.813 

2.115 

2.417 

2.719 

3.021 

3.323 

3.625 

38.1 

1.312 

1.575 

1.837 

2.100 

2.362 

2.625 

2.887 

3.150 

33.2 

1.506 

1.807 

2.108 

2.410 

2.711 

3.012 

3.313 

3.614 

38.2 

1.309 

1.571 

1.832 

2.094 

2.356 

2.618 

2.880 

3.141 

33.3 

1.502 

1.802 

2.102 

2.402 

2.703 

3.003 

3.303 

3.604 

38.3 

1.305 

1.567 

1.828 

2.089 

2.350 

2.611 

2.872 

3.133 

33.4 

1.497 

1.796 

2.096 

2.395 

2.695 

2.994 

3.293 

3.593 

38.4 

1.302 

1.563 

1.823 

2.083 

2.344 

2.604 

2.865 

3.125 

33.5 

1.493 

1.791 

2.090 

2.388 

2.687 

2.985 

3.284 

3.582 

38.5 

1.299 

1.558 

1.818 

2.078 

2.338 

2.597 

2.857 

3.117 

33.6 

1.488 

1.786 

2.083 

2.381 

2.679 

2.976 

3.274 

3.571 

38.6 

1.295 

1.554 

1.813 

2.073 

2.332 

2.591 

2.850 

3.109 

33.7 

1.484 

1.780 

2.077 

2.374 

2.671 

2.967 

3.264 

3.561 

38.7 

1.292 

1.550 

1.809 

2.067 

2.326 

2.584 

2.842 

3.101 

33.8 

1.479 

1.775 

2.071 

2.367 

2.663 

2.959 

3.254 

3.550 

38.8 

1.289 

1.546 

1.804 

2.062 

2.320 

2.577 

2.835 

3.093 

33.9 

1.475 

1.770 

2.065 

2.360 

2.655 

2.950 

3.245 

3.540 

38.9 

1.285 

1.542 

1.799 

2.057 

2.314 

2.571 

2.828 

3.085 

34.0 

1.471 

1.765 

2.059 

2.353 

2.647 

2.941 

3.235 

3.529 

39.0 

1.282 

1.538 

1.795 

2.051 

2.308 

2.564 

2.821 

3.077 

34.1 

1.466 

1.760 

2.053 

2.346 

2.639 

2.933 

3.226 

3.519 

39.1 

1.279 

1.535 

1.790 

2.046 

2.302 

2.558 

2.813 

3.069 

34.2 

1.462 

1.754 

2.047 

2.339 

2.632 

2.924 

3.216 

3.509 

39.2 

1.276 

1.531 

1.786 

2.041 

2.296 

2.551 

2.806 

3.061 

34.3 

1.458 

1.749 

2.041 

2.332 

2.624 

2.915 

3.207 

3.499 

39.3 

1.272 

1.527 

1.781 

2.036 

2.290 

2.545 

2.799 

3.053 

34.4 

1.453 

1.744 

2.035 

2.326 

2.616 

2.907 

3.198 

3.488 

39.4 

1.269 

1.523 

1.777 

2.030 

2.284 

2.538 

2.792 

3.046 

34.5 

1.449 

1.739 

2.029 

2.319 

2.609 

2.899 

3.188 

3.478 

39.5 

1.266 

1.519 

1.772 

2.025 

2.278 

2.532 

2.785 

3.038 

34.6 

1.445 

1.734 

2.023 

2.312 

2.601 

2.890 

3.179 

3.468 

39.6 

1.263 

1.515 

1.768 

2.020 

2.273 

2.525 

2.778 

3.030 

34.7 

1.441 

1.729 

2.017 

2.305 

2.594 

2.882 

3.170 

3.458 

39.7 

1.259 

1.511 

1.763 

2.015 

2.267 

2.519 

2.771 

3.023 

34.8 

1.437 

1.724 

2.011 

2.299 

2.586 

2.874 

3.161 

3.448 

39.8 

1.256 

1.508 

1.759 

2.010 

2.261 

2.513 

2.764 

3.015 

34.9 

1.433 

1.719 

2.006|  2.292 

2.579 

2.865 

3.152 

3.438 

39.9 

1.253 

1.504 

1.754 

2.005 

2.256 

2.506 

2.757 

3.008 

235 

TABLE    XXVIII  — CONTINUED. 

TABLE    OF    VELOCITIES    OF    TUBES   IN   MEASURING    FLUMES,   IN    FEET   PER    SECOND.       THE    TIME 

OCCUPIED   IN  PASSING   FROM   THE   UPSTREAM   TO   THE    DOWNSTREAM   TRANSIT 

STATION,  AND  THE   DISTANCE  BETWEEN   THEM,  BEING  GIVEN. 


TIME 
See's. 

DISTANCE  BETWEEN  THE  TRANSIT  STATIONS,  IN  FEET. 

TIME. 

See's. 

DISTANCE   BETWEEN  THE  TRANSIT  STATIONS,  IN  FEET. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

40.0 

1.250 

1.500 

1.750 

2.000 

2.250 

2.500 

2.750 

3.000 

45.0 

1.111 

1.333 

1.556 

1.778 

2.000 

2.222 

2.444 

2.667 

40.1 

1.247 

1.496 

1.746 

1.995 

2.244 

2.494 

2.743 

2.993 

45.1 

1.109 

1.330 

1.552 

1.774 

1.996 

2.217 

2.439 

2.661 

40.2 

1.244 

1.493 

1.741 

1.990 

2.239 

2.488 

2.736 

2.985 

45.2 

1.106 

1.327 

1.549 

1.770 

1.991 

2.212 

2.434 

2.655 

40.3 

1.241 

1.489 

1.737 

1.985 

2.233 

2.481 

2.730 

2.978 

45.3 

1.104 

1.325 

1.545 

1.766 

1.987 

2.208 

2.428 

2.649 

40.4 

1.238 

1.485 

1.733 

1.980 

2.228 

2.475 

2.723 

2.970 

45.4 

1.101 

1.322 

1.542 

1.762 

1.982 

2.203 

2.423 

2.643 

40.5 

1.235 

1.481 

1.728 

1.975 

2.222 

2.469 

2.716 

2.963 

45.5 

1.099 

1.319 

1.538 

1.758 

1.978 

2.198 

2.418 

2.637 

40.6 

1.232 

1.478 

1.724 

1.970 

2.217 

2.463 

2.709 

2.956 

45.6 

1.096 

1.316 

1.535 

1.754 

1.974 

2.193 

2.412 

2.632 

40.7 

1.229 

1.474 

1.720 

1.966 

2.211 

2.457 

2.703 

2.948 

45.7 

1.094 

1.313 

1.532 

1.751 

1.969 

2.188 

2.407 

2.626 

40.8 

1.225 

1.471 

1.716 

1.961 

2.206 

2.451 

2.696 

2.941 

45.8 

1.092 

1.310 

1.528 

1.747 

1.965 

2.183 

2.402 

2.620 

40.9 

1.222 

1.467 

1.711 

1.956 

2.200 

2.445 

2.689 

2.934 

45.9 

1.089 

1.307 

1.525 

1.743 

1.961 

2.179 

2.397 

2.614 

41.0 

1.220 

1.463 

1.707 

1.951 

2.195 

2.439 

2.683 

2.927 

46.0 

1.087 

1.304 

1.522 

1.739 

1.957 

2.174 

2.391 

2.609 

41.1 

1.217 

1.460 

1.703 

1.946 

2.190 

2.433 

2.676 

2.920 

46.1 

1.085 

1.302 

1.518 

1.735 

1.952 

2.169 

2.386 

2.603 

41.2 

1.214 

1.456 

1.699 

1.942 

2.184 

2.427 

2.670 

2.913 

46.2 

1.082 

1.299 

1.515 

1.732 

1.948 

2.165 

2.381 

2.597 

41.3 

1.211 

1.453 

1.695 

1.937 

2.179 

2.421 

2.663 

2.906 

46.3 

1.080 

1.296 

1.512 

1.728 

1.944 

2.160 

2.376 

2.592 

41.4 

1.208 

1.449 

1.691 

1.932 

2.174 

2.415 

2.657 

2.899 

46.4 

1.078 

1.293 

1.509 

1.724 

1.940 

2.155 

2.371 

2.586 

41.5 

1.205 

1.446 

1.687 

1.928 

2.169 

2.410 

2.651 

2.892 

46.5 

1.075 

1.290 

1.505 

1.720 

1.935 

2.151 

2.366 

2.581 

41.6 

1.202 

1.442 

1.683 

1.923 

2.163 

2.404 

2.644 

2.885 

46.6 

1.073 

1.288 

1.502 

1.717 

1.931 

2.146 

2.361 

2.575 

41.7 

1.199 

1.439 

1.679 

1.918 

2.158 

2.398 

2.638 

2.878 

46.7 

1.071 

1.285 

1.499 

1.713 

1.927 

;2.141 

2.355 

2.570 

41.8 

1.196 

1.435 

1.675 

1.914 

2.153 

2.392 

2.632 

2.871 

46.8 

1.068 

1.282 

1.496 

1.709 

1.923 

2.137 

2.350 

2.564 

41.9 

1.193 

1.432 

1.671 

1.909 

2.148 

2.387 

2.625 

2.864 

46.9 

1.066 

1.279 

1.493 

1.706 

1.919 

2.132 

2.345 

2.559 

42.0 

1.190 

1.429 

1.667 

1.905 

2.143 

2.381 

2.619 

2.857 

47.0 

1.064 

1.277 

1.489 

1.702 

1.915 

2.128 

2.340 

2.553 

42.1 

1.188 

1.425 

1.663 

1.900 

2.138 

2.375 

2.613 

2.850 

47.1 

1.062 

1.274 

1.486 

1.699 

1.911 

2.123 

2.335 

2.548 

42.2 

1.185 

1.422 

1.659 

1.896 

2.133 

2.370 

2.607 

2.844 

47.2 

1.059 

1.271 

1.483 

1.695 

1.907 

2.119 

2.331 

2.542 

42.3 

1.182 

1.418 

1.655 

1.891 

2.128 

2.364 

2.600 

2.837 

47.3 

1.057 

1.268 

1.480 

1.691 

1.903 

2.114 

2.326 

2.537 

42.4 

1.179 

1.415 

1.651 

1.887 

2.123 

2.358 

2.594 

2.830 

47.4 

1.055 

1.266 

1.477 

1.688 

1.899 

2.110 

2.321 

2.532 

42.5 

1.176 

1.412 

1.647 

1.882 

2.118 

2.353 

2.588 

2.824 

47.5 

1.053 

1.263 

1.474 

1.684 

1.895 

2.105 

2.316 

2.526 

42.6 

1.174 

1.408 

1.643 

1.878 

2.113 

2.347 

2.582 

2.817 

47.6 

1.050 

1.261 

1.471 

1.681 

1.891 

2.101 

2.311 

2.521 

42.7 

1.171 

1.405 

1.639 

1.874 

2.108 

2.342 

2.576 

2.810 

47.7 

1.048 

1.258 

1.468 

1.677 

1.887 

2.096 

2.306 

2.516 

42.8 

1.168 

1.402 

1.636 

1.869 

2.103 

2.336 

2.570 

2.804 

47.8 

1.046 

1.255 

1.464 

1.674 

1.883 

2.092 

2.301 

2.510 

42.9 

1.166 

1.399 

1.632 

1.865 

2.098 

2.331 

2.564 

2.797 

47.9 

1.044 

1.253 

1.461 

1.670 

1.879 

2.088 

2.296 

2.505 

43.0 

1.163 

1.395 

1.628 

1.860 

2.093 

2.326 

2.558 

2.791 

48.0 

1.042 

1.250 

1.458 

1.667 

1.875 

2.083 

2.292 

2.500 

43.1 

1.160 

1.392 

1.624 

1.856 

2.088 

2.320 

2.552 

2.784 

48.1 

1.040 

1.247 

1.455 

1.663 

1.871 

2.079 

2.287 

2.495 

43.2 

1.157 

1.389 

1.620 

1.852 

2.083 

2.315 

2.546 

2.778 

48.2 

3.037 

1.245 

1.452 

1.660 

1.867 

2.075 

2.282 

2.490 

43.3 

1.155 

1.386 

1.617 

1.848 

2.079 

2.309 

2.540 

2.771 

48.3 

1.035 

1.242 

1.449 

1.656 

1.863 

2.070 

2.277 

2.484 

43.4 

1.152 

1.382 

1.613 

1.843 

2.074 

2.304 

2.535 

2.765 

48.4 

1.033 

1.240 

1.446 

1.653 

1.860 

2.066 

2.273 

2.479 

43.5 

1.149 

1.379 

1.609 

1.839 

2.069 

2.299 

2.529 

2.759 

48.5 

1.031 

1.237 

1.443 

1.649 

1.856 

2.062 

2.268 

2.474 

43.6 

1.147 

1.376 

1.606 

1.835 

2.064 

2.294 

2.523 

2.752 

48.6 

1.029 

1.235 

1.440 

1.646 

1.852 

2.058 

2.263 

2.469 

43.7 

1.144 

1.373 

1.602 

1.831 

2.059 

2.288 

2.517 

2.746 

48.7 

1.027 

1.232 

1.437 

1.643 

1.848 

2.053 

2.259 

2.464 

43.8 

1.142 

1.370 

1.598 

1.826 

2.055 

2.283 

2.511 

2.740 

48.8 

1.025 

1.230 

1.434 

1.639 

1.844 

2.049 

2.254 

2.459 

43.9 

1.139 

1.367 

1.595 

1.822 

2.050 

2.278 

2.506 

2.733 

48.9 

1.022 

1.227 

1.431 

1.636 

1.840 

2.045 

2.249 

2.454 

44.0 

1.136 

1.364 

1.591 

1.818 

2.045 

2.273 

2.500 

2.727 

49.0 

1.020 

1.224 

1.429 

1.633 

1.837 

2.041 

2.245 

2.449 

44.1 

1.134 

1.361 

1.587 

1.814 

2.041 

2.268 

2.494 

2.721 

49.1 

1.018 

1.222 

1.426 

1.629 

1.833 

2.037 

2.240 

2.444 

44.2 

1.131 

1.357 

1.584 

1.810 

2.036 

2.262 

2.489 

2.715 

49.2 

1.016 

1.220 

1.423 

1.626 

1.829 

2.033 

2.236 

2.439 

44.3 

1.129 

1.354 

1.580 

1.806 

2.032 

2.257 

2.483 

2.709 

49.3 

1.014 

1.217 

1.420 

1.623 

1.826 

2.028 

2.231 

2.434 

44.4 

1.126 

1.351 

1.577 

1.802 

2.027 

2.252 

2.477 

2.703 

49.4 

1.012 

1.215 

1.417 

1.619 

1.822 

2.024 

2.227 

2.429 

44.5 

1.124 

1.348 

1.573 

1.798 

2.022 

2.247 

2.472 

2.697 

49.5 

1.010 

1.212 

1.414 

1.616 

1.818 

2.020 

2.222 

2.424 

44.6 

1.121 

1.345 

1.570 

1.794 

2.018 

2.242 

2.466 

2.691 

49.6 

1.008 

1.210 

1.411 

1.613 

1.815 

2.016 

2.218 

2.419 

44.7 

1.119 

1.342 

1.566 

1.790 

2.013 

2.237 

2.461 

2.685 

49.7 

1.006 

1.207 

1.408 

1.610 

1.811 

2.012 

2.213 

2.414 

44.8  1.116  1.339 

1.562 

1.786 

2.009 

2.232 

2.455 

2.679 

49.8 

1.004 

1.205 

1.406 

1.606 

1.807 

2.008 

2.209 

2.410 

44.9  1.114]  1.336 

1.559 

1.782 

2.004 

2.227 

2.450 

2.673 

49.9 

1.002 

1.202 

1.403 

1.603 

1.804 

2.004 

2.204 

2.405 

236 
TABLE    XXVIII  —  CONTINUED. 

TABLE    OF    VELOCITIES    OF    TUBES    IN  MEASURING    FLUMES,    IN    FEET    PER    SECOND.       THE    TIME 

OCCUPIED  IN  PASSING   FROM   THE   UPSTREAM   TO   THE    DOWNSTREAM   TRANSIT 

STATION,  AND  THE   DISTANCE  BETWEEN   THEM,  BEING  GIVEN. 


TIME 

See's. 

DISTANCE  BETWEEN  THE  TRANSIT  STATIONS,  IN  FEET. 

TIME 

See's. 

DISTANCE  BEHVEEN  TIIE  TRANSIT  STATIONS,  IN  FEET. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

50.0 

1.000 

1.200 

1.400 

1.60( 

1.800 

2.000 

2.200 

2.400 

55.0 

0.909 

1.091 

1.273 

1.455 

1.636 

1.818 

2.000 

2.182 

50.1 

0.998 

1.198 

1.397 

1.597 

1.796 

1.996 

2.196 

2.395 

55.1 

0.907 

1.089 

1.270 

1.452 

1.633 

1.815 

1.996 

2.178 

50.2 

0.99G 

1.195 

1.394 

1.594 

1.793 

1.992 

2.191 

2.390 

55.2 

0.906 

•1.087 

1.268 

1.449 

1.630 

1.812 

1.993 

2.174 

50.3 

0.994 

1.193 

1.392 

1.590 

1.789 

1.988 

2.187 

2.386 

55.3 

0.904 

1.085 

1.266 

1.447 

1.627 

1.808 

1.989 

2.170 

50.4 

0.992 

1.190 

1.389 

1.587 

1.786 

1.984 

2.183 

2.381 

55.4 

0.903 

1.083 

1.264 

1.444 

1.625 

1.805 

1.986 

2.166 

50.5 

0.990 

1.188 

1.386 

1.584 

1.782 

1.980 

2.178 

2.376 

55.5 

0.901 

1.081 

1.261 

1.441 

1.622 

1.802 

1.982 

2.162 

50.6 

0.988 

1.186 

1.383 

1.581 

1.779 

1.976 

2.174 

2.372 

55.6 

0.899 

1.079 

1.259 

1.439 

1.619 

1.799 

1.978 

2.158 

50.7 

0.986 

1.183 

1.381 

1.578 

1.775 

1.972 

2.17C 

2.367 

55.7 

0.898 

1.077 

1.257 

1.436 

1.616 

1.795 

1.975 

2.154 

50.8 

0.984 

1.181 

1.378 

1.575 

1.772 

1.969 

2.165 

2.362 

55.8 

0.896 

1.075 

1.254 

1.434 

1.613 

1.792 

1.971 

2.151 

50.9 

0.982 

1.179 

1.375 

1.572 

1.768 

1.965 

2.161 

2.358 

55.9 

0.894 

1.073 

1.252 

1.431 

1.610 

1.789 

1.968 

2.147 

51.0 

0.980 

1.176 

1.373 

1.569 

1.765 

1.961 

2.157 

2.353 

56.0 

0.893 

1.071 

1.250 

1.429 

1.607 

1.786 

1.964 

2.143 

51.1 

0.978 

1.174 

1.370 

1.566 

1.761 

1.957 

2.153 

2.348 

56.1 

0.891 

1.070 

1.248 

1.426 

1.604 

1.783 

1.961 

2.139 

51.2 

0.977 

1.172 

1.367 

1.562 

1.758 

1.953 

2.148 

2.344 

56.2 

0.890 

1.068 

1.246 

1.423 

1.601 

1.779 

1.957 

2.135 

51.3 

0.975 

1.170 

1.365 

1.559 

1.754 

1.949 

2.144 

2.339 

56.3 

0.888 

1.066 

1.243 

1.421 

1.599 

1.776 

1.954 

2.131 

51.4 

0.973 

1.167 

1.362 

1.556 

1.751 

1.946 

2.140 

2.335 

56.4 

0.887 

1.064 

1.241 

1.418 

1.596 

1.773 

1.950 

2.128 

51.5 

0.971 

1.165 

1.359 

1.553 

1.748 

1.942 

2.136 

2.330 

56.5 

0.885 

1.062 

1.239 

1.416 

1.593 

1.770 

1.947 

2.124 

51.6 

0.969 

1.163 

1.357 

1.550 

1.744 

1.938 

2.132 

2.326 

56.6 

0.883 

1.060 

1.237 

1.413 

1.590 

1.767 

1.943 

2.120 

51.7 

0.967 

1.161 

1.354 

1.547 

1.741 

1.934 

2.128 

2.321 

56.7 

0.882 

1.058 

1.235 

1.411 

1.587 

1.764 

1.940 

2.116 

51.8 

0.965 

1.158 

1.351 

1.544 

1.737 

1.931 

2.124 

2.317 

56.8 

0.880 

1.056 

1.232 

1.408 

1.585 

1.761 

1.937 

2.113 

51.9 

0.963 

1.156 

1.349 

1.541 

1.734 

1.927 

2.119 

2.312 

56.9 

0.879 

1.054 

1.230 

1.406 

1.582 

1.757 

1.933 

2.109 

52.0 

0.962 

1.154 

1.346 

1.538 

1.731 

1.923 

2.115 

2.308 

57.0 

0.877 

1.053 

1.228 

1.404 

1.579 

1.754 

1.930 

2.105 

52.1 

0.960 

1.152 

1.344 

1.536 

1.727 

1.919 

2.111 

2.303 

57.1 

0.876 

1.051 

1.226 

1.401 

1.576 

1.751 

1.926 

2.102 

52.2 

0.958 

1.149 

1.341 

1.533 

1.724 

1.916 

2.107 

2.299 

57.2 

0.874 

1.049 

1.224 

1.399 

1.573 

1.748 

1.923 

2.098 

52.3 

0.956 

1.147 

1.338 

1.530 

1.721 

1.912 

2.103 

2.294 

57.3 

0.873 

1.047 

1.222 

1.396 

1.571 

1.745 

1.920 

2.094 

52.4 

0.954 

1.145 

1.336 

1.527 

1.718 

1.908 

2.099 

2.290 

57.4 

0.871 

1.045 

1.220 

1.394 

1.568 

1.742 

1.916 

2.091 

52.5 

0.952 

1.143 

1.333 

1.524 

1.714 

1.905 

2.095 

2.286 

57.5 

0.870 

1.043 

1.217 

1.391 

1.565 

1.739 

1.913 

2.087 

52.6 

0.951 

1.141 

1.331 

1.521 

1.711 

1.901 

2.091 

2.281 

57.6 

0.868 

1.042 

1.215 

1.389 

1.562 

1.736 

1.910 

2.083 

52.7 

0.949 

1.139 

1.328 

1.518 

1.708 

1.898 

2.087 

2.277 

57.7 

0.867 

1.040 

1.213 

1.386 

1.560 

1.733 

1.906 

2.080 

52.8 

0.947 

1.136 

1.326 

1.515 

1.705 

1.894 

2.083 

2.273 

57.8 

0.865 

1.038 

1.211 

1.384 

1.557 

1.730 

1.903 

2.076 

52.9 

0.945 

1.134 

1.323 

1.512 

1.701 

1.890 

2.079 

2.268 

57.9 

0.864 

1.036 

1.209 

1.382 

1.554 

1.727 

1.900 

2.073 

53.0 

0.943 

1.132 

1.321 

1.509 

1.698 

1.887 

2.075 

2.264 

58.0 

0.862 

1.034 

1.207 

1.379 

1.552 

1.724 

1.897 

2.069 

53.1 

0.942 

1.130 

1.318 

1.507 

1.695 

1.883 

2.072 

2.260 

58.1 

0.861 

1.033 

1.205 

1.377 

1.549 

1.721 

1.893 

2.065 

53.2 

0.940 

1.128 

1.316 

1.504 

1.692 

1.880 

2.068 

2.256 

58.2 

0.859 

1.031 

1.203 

1.375 

1.546 

1.718 

1.890 

2.062 

53.3 

0.938 

1.126 

1.313 

1.501 

1.689 

1.876 

2.064 

2.251 

58.3 

0.858 

1.029 

1.201 

1.372 

1.544 

1.715 

1.887 

2.058 

53.4 

0.936 

1.124 

1.311 

1.498 

1.685 

1.873 

2.060 

2.247 

58.4 

0.856 

1.027 

1.199 

1.370 

1.541 

1.712 

1.884 

2.055 

53.5 

0.935 

1.121 

1.308 

1.495 

1.682 

1.869 

2.056 

2.243 

58.5 

0.855 

1.026 

1.197 

1.368 

1.538 

1.709 

1.880 

2.051 

53.6 

0.933 

1.119 

1.306 

1.493 

1.679 

1.866 

2.052 

2.239 

58.6 

0.853 

1.024 

1.195 

1.365 

1.536 

1.706 

1.877 

2.048 

53.7 

0.931 

1.117 

1.304 

1.490 

1.676 

1.862 

2.048 

2.235 

58.7 

0.852 

1.022 

1.193 

1.363 

1.533 

1.704 

1.874 

2.044 

53.8 

0.929 

1.115 

1.301 

1.487 

1.673 

1.859 

2.045 

2.230 

58.8 

0.850 

1.020 

1.190 

1.361 

1.531 

1.701 

1.871 

2.041 

53.9 

0.928 

1.113 

1.299 

1.484 

1.670 

1.855 

2.041 

2.226 

58.9 

0.849 

1.019 

1.188 

1.358 

1.528 

1.698 

1.868 

2.037 

54.0 

0.926 

1.111 

1.296 

1.481 

1.667 

1.852 

2.037 

2.222 

59.0 

0.847 

1.017 

1.186 

1.356 

1.525 

1.695 

1.864 

2.034 

54.1 

0.924 

1.109 

1.294 

1.479 

1.664 

1.S4K 

2.033 

2.218 

59.1 

0.846 

1.015 

1.184 

1.354 

1.523 

1.692 

1.861 

2.030 

54.2 

0.923 

1.107 

1.292 

1.476 

1.661 

1.845 

2.030 

2.214 

59.2 

0.845 

1.014 

1.182 

1.351 

1.520 

1.689 

1.858 

2.027 

54.3 

0.921 

1.105 

1.289 

1.473 

1.657 

1.842 

2.026 

2.210 

59.3 

0.843 

1.012 

1.180 

1.349 

1.518 

1.686 

l.«55 

2.024 

54.4 

0.919 

1.103 

1.287 

1.471 

1.654 

1.838 

2.022 

2.206 

59.4 

0.842 

1.010 

1.178 

1.347 

1.515 

1.684 

1.852 

2.020 

54.5 

0.917 

1.101 

1.284 

1.468 

1.651 

1.835 

2.018 

2.202 

59.5 

0.840 

1.008 

1.176 

1.345 

1.513 

1.681 

1.849 

2.017 

54.6 

0.916 

1.099 

1.282 

1.465 

1.648 

1.832 

2.015 

2.198 

59.6 

0.839 

1.007 

1.174 

1.342 

1.510 

1.678 

1.846 

2.013 

54.7 

0.914 

1.097 

1.280 

1.463 

1.645 

1.828 

2.011 

2.194 

59.7 

0.838 

1.005 

1.173 

1.340 

1.508 

1.675 

1.843 

2.010 

54.8 

0.912 

1.095 

1.277 

1.460 

1.642 

1.825 

2.007 

2.190 

59.8 

0.836 

1.003 

1.171    1.338 

1.505 

1.672 

1.839 

2.007 

54.9 

0.911 

1.093 

1.275 

1.457 

1.639 

1.821 

2.004 

2.186 

59.9 

0.835 

1.002   1.169    1.336 

1.503  1.669 

1.836 

2.003 

237 

TABLE    XXVIII  — CONTINUED. 

TABLE    OF    VELOCITIES    OF    TUBES    IN  MEASURING    FLUMES,    IN    FEET    PER    SECOND.       THE    TIME 

OCCUPIED  IN  PASSING   FROM   THE   UPSTREAM   TO   THE    DOWNSTREAM  TRANSIT 

STATION,  AND  THE   DISTANCE  BETWEEN   THEM,  BEING  GIVEN. 


TIME. 
Sec'8. 

DISTANCE  BETWEEN  THE  TRANSIT  STATIONS,  IN  FEET 

TIME. 

See's. 

DISTANCE   BETWEEN  THE  TRANSIT  STATIONS,  IN  FEET. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

60.0 

0.833 

1.000 

1.167 

1.333 

1.500 

1.667 

1.833 

2.000 

65.0 

0.769 

0.923 

1.077 

1.231 

1.385 

1.538 

1.692 

1.846 

60.1 

0.832 

0.998 

1.165 

1.331 

1.498 

1.664 

1.830 

1.997 

65.1 

0.768 

0.922 

1.075 

1.229 

1.382 

1.536 

1.690 

1.843 

60.2 

0.831 

0.997 

1.163 

1.329 

1.495 

1.661 

1.827 

1.993 

65.2 

0.767 

0.920 

1.074 

1.227 

1.380 

1.534 

1.687 

1.840 

60.3 

0.829 

0.995 

1.161 

1.327 

1.493 

1.658 

1.824 

1.990 

65.3 

0.766 

0.919 

1.072 

1.225 

1.378 

1.531 

1.685 

1.838 

60.4 

0.828 

0.993 

1.159 

1.325 

1.490 

1.656 

1.821 

1.987 

65.4 

0.765 

0.917 

1.070 

1.223 

1.376 

1.529 

1.682 

1.835 

60.5 

0.826 

0.992 

1.157 

1.322 

1.488 

1.653 

1.818 

1.983 

65.5 

0.763 

0.916 

1.069 

1.221 

1.374 

1.527 

.679 

1.832 

60.6 

0.825 

0.990 

1.155 

1.320 

1.485 

1.650 

1.815 

1.980 

65.6 

0.762 

0.915 

1.067 

1.220 

1.372 

1.524 

.677 

1.829 

60.7 

0:824 

0.988 

1.153 

1.318 

1.483 

1.647 

1.812 

1.977 

65.7 

0.761 

0.913 

1.065 

1.218 

1.370 

1.522 

.674 

1.826 

60.8 

0.822 

0.987 

1.151 

1.316 

1.480 

1.645 

1.809 

1.974 

65.8 

0.760 

0.912 

1.064 

1.216 

1.368 

1.520 

.672 

1.824 

60.9 

0.821 

0.985 

1.149 

1.314 

1.478 

1.642 

1.806 

1.970 

65:9 

0.759 

0.910 

1.062 

1.214 

1.366 

1.517 

.669 

1.821 

61.0 

0.820 

0.984 

1.148 

1.311 

1.475 

1.639 

1.803 

1.967 

66.0 

0.758 

0.909 

1.061 

1.212 

1.364 

1.515 

1.667 

1.818 

61.1 

0.818 

0.982 

1.146 

1.309 

1.473 

1.637 

1.800 

1.964 

66.1 

0.756 

0.908 

1.059 

1.210 

1.362 

1.513 

1.664 

1.815 

61.2 

0.817 

0.980 

1.144 

1.307 

1.471 

1.634 

1.797 

1.961 

66.2 

0.755 

0.906 

1.057 

1.208 

1.360 

1.511 

1.662 

1.813 

61.3 

0.816 

0.979 

L142 

1.305 

1.468 

1.631 

1.794 

1.958 

66.3 

0.754 

0.905 

1.056 

1.207 

1.357 

1.508 

1.659 

1.810 

61.4 

0.814 

0.977 

1.140 

1.303 

1.466 

1.629 

1.792 

1.954 

66.4 

0.753 

0.904 

1.054 

1.205 

1.355 

1.506 

1.657 

1.807 

61.5 

0.813 

0.976 

1.138 

1.301 

1.463 

1.626 

1.789 

1.951 

66.5 

0.752 

0.902 

1.053 

1.203 

1.353 

1.504 

.654 

1.805 

61.6 

0.812 

0.974 

1.136 

1.299 

1.461 

1.623 

1.786 

1.948 

66.6 

0.751 

0.901 

1.051 

1.201 

1.351 

1.502 

.652 

1.802 

61.7 

0.810 

0.972 

1.135 

1.297 

1.459 

1.621 

1.783 

1.945 

66.7 

0.750 

0.900 

1.049 

1.199 

1.349 

1.499 

.649 

1.799 

61.8 

0.809 

0.971 

1.133 

1.294 

1.456 

1.618 

1.780 

1.942 

66.8 

0.749 

0.898 

1.048 

1.198 

1.347 

1.497 

.647 

1.796 

61.9 

0.808 

0.969 

1.131 

1.292 

1.454 

1.616 

1.777 

1.939 

66.9 

0.747 

0.897 

1.046 

1.196 

1.345 

1.495 

.644 

1.794 

62.0 

0.806 

0.968 

1.129 

1.290 

1.452 

1.613 

1.774 

1.935 

67.0 

0.746 

0.896 

1.045 

1.194 

1.343 

1.493 

.642 

1.791 

62.1 

0.805 

0.966 

1.127 

1.288 

1.449 

1.610 

1.771 

1.932 

67.1 

0.745 

0.894 

1.043 

1.192 

1.341 

1.490 

.639 

1.788 

62.2 

0.804 

0.965 

1.125 

1.286 

1.447 

1.608 

1.768 

1.929 

67.2 

0.741 

0.893 

1.042 

1.190 

1.339 

1.488 

1.637 

1.786 

62.3 

0.803 

0.963 

1.124 

1.284 

1.445 

1.605 

1.766 

1.926 

67.3 

0.743 

0.892 

1.040 

1.189 

1.337 

1.486 

1.634 

1.783 

62.4 

0.801 

0.962 

1.122 

1.282 

1.442 

1.603 

1.763 

1.923 

67.4 

0.742 

0.890 

1.039 

1.187 

1.335 

1.484 

1.632 

1.780 

62.5 

0.800 

0.960 

1.120 

1.280 

1.440 

1.600 

1.760 

1.920 

67.5 

0.741 

0.839 

1.037 

1.185 

1.333 

1.481 

1.630 

1.778 

62.6 

0.799 

0.958 

1.118 

1.278 

1.438 

1.597 

1.757 

1.917 

67.6 

0.740 

0.888 

1.036 

1.183 

1.331 

1.479 

1.627 

1.775 

62.7 

0.797 

0.957 

1.116 

1.276 

1.435 

1.595 

1.754 

1.914 

67.7 

0.739 

0.886 

1.034 

1.182 

1.329 

1.477 

1.625 

1.773 

62.8 

0.796 

0.955 

1.115 

1.274 

1.433 

1.592 

1.752 

1.911 

67.8 

0.737 

0.885 

1.032 

1.180 

1.327 

1.475 

1.622 

1.770 

62.9 

0.795 

0.954 

1.113 

1.272 

1.431 

1.590 

1.749 

1.908 

67.9 

0.736 

0.884 

1.031 

1.178 

1.325 

1.473 

1.620 

1.767 

63.0 

0.794 

0.952 

1.111 

1.270 

1.429 

1.587 

1.746 

1.905 

68.0 

0.735 

0.882 

1.029 

1.176 

1.324 

1.471 

1.618 

1.765 

63.1 

0.792 

0.951 

1.109 

1.268 

1.426 

1.585 

1.743 

1.902 

68.1 

0.734 

0.881 

1.028 

1.175 

1.322 

1.468 

1.615 

1.762 

63.2 

0.791 

0.949 

1.108 

1.266 

1.424 

1.582 

1.741 

1.899 

68.2 

0.733 

0.880 

1.026 

1.173 

1.320 

1.466 

1.613 

1.760 

63.3 

0.790 

0.948 

1.106 

1.264 

1.422 

1.580 

1.738 

1.896 

68.3 

0.732 

0.878 

1.025 

1.171 

1.318 

1.464 

1.611 

1.757 

63.4 

0.789 

0.946 

1.104 

1.262 

1.420 

1.577 

1.735 

1.893 

68.4 

0.731 

0.877 

1.023 

1.170 

1.316 

1.462 

1.608 

1.754 

63.5 

0.787 

0.945 

1.102 

1.260 

1.417 

1.575 

1.732 

1.890 

68.5 

0.730 

0.876 

1.022 

1.168 

1.314 

1.460 

1.606 

1.752 

63.6 

0.786 

0.943 

1.101 

1.258 

1.415 

1.572 

1.730 

1.887 

68.6 

0.729 

0.875 

1.020 

1.166 

1.312 

1.458 

1.603 

1.749 

63.7 

0.785 

0.942 

1.099 

1.256 

1.413 

1.570 

1.727 

1.884 

68.7 

0.728 

0.873 

1.019 

1.164 

1.310 

1.456 

1.601 

1.747 

63.8 

0.784 

0.940 

1.097 

1.254 

1.411 

1.567 

1.721 

1.881 

68.8 

0.727 

0.872 

1.017 

1.163 

1.308 

1.453 

1.599 

1.744 

63.9 

0.782 

0.939 

1.095 

1.252 

1.408 

1.565 

1.721 

1.878 

68.9 

0.726 

0.871 

1.016 

1.161 

1.306 

1.451 

1.597 

1.742 

64.0 

0.781 

0.937 

1.094 

1.250 

1.406 

1.562 

1.719 

1.875 

69.0 

0.725 

0.870 

1.014 

1.159 

1.304 

1.449 

1.594 

1.739 

64.1 

0.780 

0.936 

1.092 

1.248 

1.404 

1.560 

1.716 

1.872 

69.1 

0.724 

0.868 

1.013 

1.158 

1.302 

1.447 

1.592 

1.737 

64.2 

0.779 

0.935 

1.090 

1.-246 

1.402 

1.553 

1.713 

1.869 

69.2 

0.723 

0.867 

1.012 

1.156 

1.301 

1.445 

1.590 

1.734 

64.3 

0.778 

0.933 

1.089 

1.244 

1.400 

1.555 

1.711 

1.866 

69.3 

0.722 

0.866 

1.010 

1.154 

1.299 

1.443 

1.587 

1.732 

64.4 

0.776 

0.932 

1.087 

1.242 

1.398 

1.553 

1.708 

1.863 

69.4 

0.720 

0.865 

1.009 

1.153 

1.297 

1.441 

1.585 

1.729 

64.5 

0.775 

0.930 

1.085 

1.240 

1.395 

1.550 

1.705 

1.860 

69.5 

0.719 

0.863 

1.007 

1.151 

1.295 

1.439 

1.583 

1.727 

64.6 

0.774 

0.929 

1.084 

1.238 

1.393 

1.548 

1.703 

1.858 

69.6 

0.718 

0.862 

l.OOC 

1.149 

1.293 

1.437 

1.580 

1.724 

64.7 

0.773 

0.927 

1.082 

1.236 

1.391 

1.546 

1.700 

1.855 

69.7 

0.717 

0.861 

1.004 

1.148  1.291 

1.435 

1.578 

1.722 

C,  1,S 

0.772 

0.926 

1.080 

1.235 

1.389 

1.543 

1.698 

1.852 

69.8 

0.716 

0.860 

1.003 

1.146 

1.289 

1.433 

1.576 

1.719 

64.9  0.770 

0.924 

1.079 

1.233 

1.387 

1.541 

1.695 

1.849 

69.9 

0.715 

0.858 

1.001 

1.144 

1.288 

1.431 

1.574 

1.717 

238 
TABLE    XXVIII  — CONTINUED. 

TABLE    OF    VELOCITIES    OF    TUBES    IN  MEASURING    FLUMES,    IN    FEET    PER    SECOND.       THE    TIME 

OCCUPIED   IN  PASSING   FROM   THE   UPSTREAM   TO   THE    DOWNSTREAM   TRANSIT 

STATION,  AND  THE   DISTANCE  BETWEEN   THEM,  BEING  GIVEN. 


TIME 

Sec'B. 

DISTANCE  BETWEEN  TILE  TRANSIT  STATIONS,  IN  FEET. 

TIME. 

See's. 

DISTANCE  BETWEEN  TIIE  TRANSIT  STATIONS,  IN  FEET. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

70.0 

0.714 

0.857 

1.000 

1.143 

1.286 

1.429 

1.571 

1.714 

75.0 

0.667 

0.800 

0.933 

1.067 

1.200 

1.333 

1.467 

1.600 

70.1 

0.713 

0.850 

0.999 

1.141 

1.284 

1.427 

1.569 

1.712 

75.1 

0.666 

0.799 

0.932 

1.065 

1.198 

1.332 

1.465 

1.598 

70.2 

0.712 

0.855 

0.997 

1.140 

1.282 

1.425 

1.567 

1.709 

75.2 

0.665 

0.798 

0.931 

1.064 

1.197 

1.330 

1.463 

1.596 

70.3 

0.711 

0.853 

0.996 

1.138 

1.280 

1.422 

1.565 

1.707 

75.3 

0.664 

0.797 

0.930 

1.062 

1.195 

1.328 

1.461 

1.594 

70.4 

0.710 

0.852 

0.994 

1.136 

1.278 

1.420 

1.562 

1.705 

75.4 

0.663 

0.796 

0.928 

1.061 

1.194 

1.326 

1.459 

1.592 

70.5 

0.709 

0.851 

0.993 

1.135 

1.277 

1.418 

1.560 

1.702 

75.5 

0.662 

0.795 

0.927 

1.060 

1.192 

1.325 

1.457 

1.589  : 

70.6 

0.708 

0.850 

0.992 

1.133 

1.275 

1.416 

1.558 

1.700 

75.6 

0.661 

0.794 

0.926 

1.058 

1.190 

1.323 

1.455 

1.587  , 

70.7 

0.707 

0.849 

0.990 

1.132 

1.273 

1.414 

1.556 

1.697 

75.7 

0.661 

0.793 

0.925 

1.057 

1.189 

1.321 

1.453 

1.585 

70.8 

0.706 

0.847 

0.989 

1.130 

1.271 

1.412 

1.554 

1.695 

75.8 

0.660 

0.792 

0.923 

1.055 

1.187 

1.319 

1.451 

1.583 

70.9 

0.705 

0.846 

0.987 

1.128 

1.269 

1.410 

1.551 

1.693 

75.9 

0.659 

0.791 

0.922 

1.054 

1.186 

1.318 

1.449 

1.581 

71.0 

0.704 

0.845 

0.986 

1.127 

1.268 

1.408 

1.549 

1.690 

76.0 

0.658 

0.789 

0.921 

1.053 

1.184 

1.316 

1.447 

1.579 

71.1 

0.703 

0.844 

0.985 

1.125 

1.266 

1.406 

1.547 

1.688 

76.1 

0.657 

0.788 

0.920 

1.051 

1.183 

1.314 

1.445 

1.577 

71.2 

0.702 

0.843 

0.983 

1.124 

1.264 

1.404 

1.545 

1.685 

76.2 

0.656 

0.787 

0.919 

1.050 

1.181 

1.312 

1.444 

1.575 

71.3 

0.701 

0.842 

0.982 

1.122 

1.262 

1.403 

1.543 

1.683 

76.3 

0.655 

0.786 

0.917 

1.048 

1.180 

1.311 

1.442 

1.573 

71.4 

0.700 

0.840 

0.980 

1.120 

1.261 

1.401 

1.541 

1.681 

76.4 

0.654 

0.785 

0.916 

1.047 

1.178 

1.309 

1.440 

1.571 

71.5 

O.G99 

0.839 

0.979 

1.119 

1.259 

1.399 

1.538 

1.678 

76.5 

0.654 

0.784 

0.915 

1.046 

1.176 

1.307 

1.438 

1.569 

71.6 

0.698 

0.838 

0.978 

1.117 

J.257 

1.397 

1.536 

1.676 

76.6 

0.653 

0.783 

0.914 

1.044 

1.175 

1.305 

1.436 

1.567 

71.7 

0.697 

0.837 

0.976 

1.116 

1.255 

1.395 

1.534 

1.674 

76.7 

0.652 

0.782 

0.913 

1.043 

1.173 

1.304 

1.434 

1.565 

71.8 

0.696 

0.836 

0.975 

1.114 

1.253 

1.393 

1.532 

1.671 

76.8 

0.651 

0.781 

0.911 

1.042 

1.172 

1.302 

1.432 

1.562 

71.9 

0.695 

0.834 

0.974 

1.113 

1.252 

1.391 

1.530 

1.669 

76.9 

0.650 

0.780 

0.910 

1.040 

1.170 

1.300 

1.430 

1.560 

72.0 

O.G94 

0.833 

0.972 

1.111 

1.250 

1.389 

1.528 

1.667 

77.0 

0.649 

0.779 

0.909 

1.039 

1.169 

1.299 

1.429 

1.558 

72.1 

0.693 

0.832 

0.971 

1.110 

.248 

1.387 

1.526 

1.664 

77.1 

0.649 

0.778 

0.908 

1.038 

1.167 

1.297 

1.427 

1.556- 

72.2 

0.693 

0.831 

0.970 

1.108 

.247 

1.385 

1.524 

1.662 

77.2 

0.648 

0.777 

0.907 

1.036 

1.166 

1.295 

1.425 

1.554 

72.3 

0.692 

0.830 

0.968 

1.107 

.245 

1.383 

1.521 

1.660 

77.3 

0.647 

0.776 

0.906 

1.035 

1.164 

1.294 

1.423 

1.552 

72.4 

0.691 

0.829 

0.967 

1.105 

.243 

1.381 

1.519 

1.657 

77.4 

0.646 

0.775 

0.904 

1.034 

1.163 

1.292 

1.421 

1.550 

72.5 

0.610 

0.828 

0.966 

1.103 

.241 

.379 

1.517 

1.655 

77.5 

0.645 

0.774 

0.903 

.032 

1.161 

1.290 

1.419 

1.548 

72.6 

0.689 

0.826 

0.964 

1.102 

.240 

.377 

1.515 

1.653 

77.6 

0.644 

0.773 

0.902 

.031 

1.160 

1.289 

1.418 

1.546 

72.7 

0.688 

0.825 

0.963 

1.100 

.238 

.376 

1.513 

1.651 

77.7 

0.644 

0.772 

0.901 

.030 

1.158 

1.287 

1.416 

1.544 

72.8 

0.687 

0.824 

0.962 

1.099 

.236 

.374 

1.511 

1.648 

77.8 

0.643 

0.771 

0.900 

.028 

1.157 

1.285 

1.414 

1.542 

72.9 

0.686 

0.823 

0.960 

1.097 

1.235 

.372 

1.509 

1.646 

77.9 

0.612 

0.770 

0.899 

.027 

1.155 

1.284 

1.412 

1.540 

73.0 

0.685 

0.822 

0.959 

1.096 

1.233 

.370 

1.507 

1.614 

78.0 

0.641 

0.769 

0.897 

1.026 

1.154 

1.282 

1.410 

1.538 

73.1 

0.684 

0.821 

0.958 

1.094 

1.231 

.368 

1.505 

1.612 

78.1 

0.640 

0.768 

0.896 

1.024 

1.152 

1.280 

1.408 

1.536 

73.2 

0.683 

0.820 

0.956 

1.093 

1.230 

.366 

1.503 

1.639 

78.2 

0.639 

0.767 

0.895 

1.023 

1.151 

1.279 

1.407 

1.535 

73.3 

0.682 

0.819 

0.955 

1.091 

1.228 

.364 

1.501 

1.637 

78.3 

0.639 

0.766 

0.894 

1.022 

1.149 

1.277 

1.405 

1.533 

73.4 

0.681 

0.817 

0.954 

1.090 

1.226 

1.362 

1.499 

1.635 

78.4 

0.638 

0.765 

0.893 

1.020 

1.148 

1.276 

1.403 

1.531 

73.5 

0.680 

0.816 

0.952 

1.088 

1.224 

1.361 

1.497 

1.633 

78.5 

0.637 

0.764 

0.892 

1.019 

1.146 

1.274 

1.401 

1.529 

73.6 

0.679 

0.815 

0.951 

1.087 

1.223 

1.359 

1.495 

1.630 

78.6 

0.636 

0.763 

0.891 

1.018 

1.145 

1.272 

1.399 

1.527 

73.7 

0.678 

0.814 

0.950 

.085 

1.221 

1.357 

1.493 

1.628 

78.7 

0.635 

0.762 

0.889 

1.017 

1.144 

1.271 

1.398 

1.525 

73.8 

0.678 

0.813 

0.949 

.084 

1.220 

.355 

1.491 

1.626 

78.8 

0.635 

0.761 

0.888 

1.015 

1.142 

1.269 

1.396 

1.523 

73.9 

0.677 

0.812 

0.947 

.083 

1.218 

.353 

1.488 

1.624 

78.9 

0.634 

0.760 

0.887 

1.014 

1.141 

1.267 

1.394 

1.521 

74.0 

0.676 

0.811 

0.946 

.081 

.216 

.351 

1.486 

1.622 

79.0 

0.633 

0.759 

0.886 

1.013 

1.139 

1.266 

1.392 

1.519 

74.1 

0.675 

0.810 

0.945 

.080 

.215 

.35(1 

1.484 

1.619 

79.1 

0.632 

0.759 

0.885 

1.011 

1.138 

1.264 

1.391 

1.517 

74.2 

0.674 

0.809 

0.943 

.078 

.213 

.34S 

1.482 

1.617 

79.2 

0.631 

0.758 

0.884 

1.010 

1.136 

1.263 

1.389 

1.515 

74.3 

0.673 

0.808 

0.942 

.077 

.211 

.34(1 

1.480 

1.615 

79.3 

0.631 

0.757 

0.883 

1.009 

1.135 

1.261 

1.387 

1.513 

74.4 

0.672 

0.806 

0.941 

.075 

.210 

.344 

1.478 

1.613 

79.4 

0.630 

0.756 

0.882 

1.008 

1.134 

1.259 

1.385 

1.511 

74.5 

0.671 

0.805 

0.940 

1.074 

1.208 

.342 

1.477 

1.611 

79.5 

0.629 

0.755 

0.881 

1.006 

1.132 

1.258 

1.384 

1.509 

74.6 

0.670 

0.804 

0.938 

1.072 

1.206 

.340 

1.475 

1.609 

79.6 

0.628 

0.754 

0.879 

1.005 

1.131 

1.256 

1.382 

1.508 

74.7 

0.669 

0.803 

0.937 

1.071 

1.205  1.339 

1.473 

1.606 

79.7 

0.627 

0.753 

0.878 

1.004 

1.129 

1.255 

1.380 

1.506 

74.8 

e.efis 

0.802  0.93G 

1.070 

1.203  1.337 

1.471 

1.604 

79.8 

0.627 

0.752 

0.877 

1.003 

1.128 

1.253 

1.378 

1.504 

74.9 

0.668 

0.801  0.935 

1.068 

1.202   1.335 

1.469 

1.602 

79.9 

0.626 

0.751 

0.876 

1.001 

1.126 

1.252   1.377 

1.502 

239 

TABLE    XXVIII— CONTINUED. 

TABLE    OF    VELOCITIES    OF    TUBES    IN  MEASURING    FLUMES,    IN    FEET   PER    SECOND.       THE    TIME 

OCCUPIED   IN  PASSING   FROM   THE   UPSTREAM   TO   THE    DOWNSTREAM  TRANSIT 

STATION,  AND  THE  DISTANCE  BETWEEN  THEM,  BEING  GIVEN. 


TIME. 

See's. 

DIS 

50. 

rANCE  BETWEEN  THE  TRANSIT  STATIONS,  IN  FEET. 

TIME. 

See's. 

DISTANCE   BETWEEN  THE  TRANSIT  STATIONS,  IN  FEET. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

80.0 

0.625 

0.750 

0.875 

1.000 

1.125 

1.250 

1.375 

1.500 

85.0 

0.588 

0.706 

0.824 

0.941 

1.059 

1.176 

1.294 

1.412 

80.1 

0.624 

0.749 

0.874 

0.999 

1.124 

1.248 

1.373 

1.498 

85.1 

0.588 

0.705 

0.823 

0.940 

1.058 

1.175 

1.293 

1.410 

80.2 

0.623 

0.748 

0.873 

0.998 

1.122 

1.247 

1.372 

1.496 

85.2 

0.587 

0.704 

0.822 

0.939 

1.056 

1.174 

1.291 

1.408 

80.3 

0.623 

0.747 

0.872 

0.996 

1.121 

1.245 

1.370 

1.494 

85.3 

0.586 

0.703 

0.821 

0.938 

1.055 

1.172 

1.290 

1.407 

80.4 

0.622 

0.746 

0.871 

0.995 

1.119 

1.244 

1.368 

1.493 

85.4 

0.585 

0.703 

0.820 

0.937 

1.054 

1.171 

1.288 

1.405 

80.5 

0.621 

0.745 

0.870 

0.994 

1.118 

1.242 

1.366 

1.491 

85.5 

0.585 

0.702 

0.819 

0.936 

1.053 

1.170 

1.287 

1.404 

80.6 

0.620 

0.744 

0.868 

0.993 

1.117 

1.241 

1.365 

1.489 

85.6 

0.584 

0.701 

0.818 

0.935 

1.051 

1.168 

1.285 

1.402 

80.7 

0.620 

0.743 

0.867 

0.991 

1.115 

1.239 

1.363 

1.487 

85.7 

0.583 

0.700 

0.817 

0.933 

1.050 

1.167 

1.284 

1.400 

80.8 

0.619 

0.743 

0.866 

0.990 

1.114 

1.238 

1.361 

1.485 

85.8 

0.583 

0.699 

0.816 

0.932 

1.049 

1.166 

1.282 

1.399 

80.9 

0.618 

0.742 

0.865 

0.989 

1.112 

1.236 

1.360 

1.483 

85.9 

0.582 

0.698 

0.815 

0.931 

1.048 

1.164 

1.281 

1.897 

81.0 

0.617 

0.741 

0.864 

0.988 

1.111 

1.235 

1.358 

1.481 

86.0 

0.581 

0.698 

0.814 

0.930 

1.047 

1.163 

1.279 

1.395 

81.1 

0.617 

0.740 

0.863 

0.986 

1.110 

1.233 

1.356 

1.480 

86.1 

0.581 

0.697 

0.813 

0.929 

1.045 

1.161 

1.278 

1.394 

81.2 

0.616 

0.739 

0.862 

0.985 

1.108 

1.232 

1.355 

1.478 

86.2 

0.580 

0.696 

0.812 

0.928 

1.044 

1.160 

1.276 

1.392 

81.3 

0.615 

0.738 

0.861 

0.984 

1.107 

1.230 

1.353 

1.476 

86.3 

0.579 

0.695 

0.811 

0.927 

1.043 

1.159 

1.275 

1.390 

81.4 

0.614 

0.737 

0.860 

0.983 

1.106 

1.229 

1.351 

1.474 

86.4 

0.579 

0.694 

0.810 

0.926 

1.042 

1.157 

1.273 

1.389 

81.5 

0.613 

0.736 

0.859 

0.982 

1.104 

1.227 

1.350 

1.472 

86.5 

0.578 

0.694 

0.809 

0.925 

1.040 

1.156 

1.272 

1.387 

81.6 

0.613 

0.735 

0.858 

0.980 

1.103 

1.225 

1.348 

1.471 

86.6 

0.577 

0.693 

0.808 

0.924 

1.039 

1.155 

1.270 

1.386 

81.7 

0.612 

0.734 

0.857 

0.979 

1.102 

1.224 

1.346 

1.469 

86.7 

0.577 

0.692 

0.807 

0.923 

1.088 

1.153 

1.269 

1.384 

81.8 

0.611 

0.733 

0.856 

0.978 

1.100 

1.222 

1.345 

1.467 

86.8 

0.576 

0.691 

0.806 

0.922 

1.037 

1.152 

1.267 

1.382 

81.9 

0.611 

0.733 

0.855 

0.977 

1.099 

1.221 

1.343 

1.465 

86.9 

0.575 

0.690 

0.806 

0.921 

1.036 

1.151 

1.266 

1.381 

82.0 

0.610 

0.732 

0.854 

0.976 

1.098 

1.220 

1.341 

1.463 

87.0 

0.575 

0.690 

0.805 

0.920 

1.034 

1.149 

1.264 

1.379 

82.1 

0.609 

0.731 

0.853 

0.974 

1.096 

1.218 

1.340 

1.462 

87.1 

0574 

0.689 

0.804 

0.918 

1.033 

1.148 

1.263 

.378 

82.2 

0.608 

0.730 

0.852 

0.973 

1.095 

1.217 

1.338 

1.460 

87.2 

0.573 

0.688 

0.803 

0.917 

1.032 

1.147 

1.261 

.376 

82.3 

0.608 

0.729 

0.851 

0.972 

1.094 

1.215 

1.337 

1.458 

87.3 

0.573 

0.687 

0.802 

0.916 

1.031 

1.145 

1.260 

.375 

82.4 

0.607 

0.728 

0.850 

0.971 

1.092 

1.214 

1.335 

1.456 

87.4 

0.572 

0.686 

0.801 

0.915 

1.030 

1.144 

1.259 

.373 

82.5 

0.606 

0.727 

0.848 

0.970 

1.091 

1.212 

1.333 

1.455 

87.5 

0.571 

0.686 

0.800 

0.914 

1.029 

1.143 

1.257 

.371 

82.6 

0.605 

0.726 

0.847 

0.969 

1.090 

1.211 

1.332 

1.453 

87.6 

0.571 

0.685 

0.799 

0.913 

1.027 

1.142 

1.256 

.370 

82.7 

0.605 

0.726 

0.846 

0.967 

1.088 

1.209 

1.330 

1.451 

87.7 

0.570 

0.684 

0.798 

0.912 

1.026 

1.140 

1.254 

1.368 

82.8 

0.604 

0.725 

0.845 

0.966 

1.087 

.208 

1.329 

1.449 

87.8 

0.569 

0.683 

0.797 

0.911 

1.025 

1.139 

1.253 

1.367 

82.9 

0.603 

0.724 

0.844 

0.965 

1.086 

1.206 

1.327 

1.448 

87.9 

0.569 

0.683 

0.796 

0.910 

1.024 

1.138 

1.251 

1.365 

83.0 

0.602 

0.723 

0.843 

0.964 

1084 

1.205 

1.325 

1.446 

88.0 

0.568 

0.632 

0.795 

0.909 

1.023 

1.136 

1.250 

1.364 

83.1 

0.602 

0.722 

0.842 

0.963 

1.083 

1.203 

1.324 

1.444 

88.1 

0.568 

0.681 

0.795 

0.908 

1.022 

1.135 

1.249 

1.362 

83.2 

0.601 

0.721 

0.841 

0.962 

1.082 

1.202 

1.322 

1.442 

88.2 

0.567 

0.680 

0.794 

0.907 

1.020 

1.134 

1.247 

1.361 

83.3 

0.600 

0.720 

0.840 

0.960 

1.080 

1.200 

1.321 

1.441 

88.3 

0.566 

0.680 

0.793 

0.906 

1.019 

1.133 

1.246 

1.359 

83.4 

0.600 

0.719 

0.839 

0.959 

l.O/'J 

1.199' 

1.319 

1.439 

88.4 

0.566 

0.679 

0.792 

0.905 

1.018 

1.131 

1.244 

1.357 

83.5 

0.599 

0.719 

0.838 

0.958 

1.078 

.198 

1.317 

1.437 

88.5 

0.565 

0.678 

0.791 

0.904 

1.017 

1.130 

1.243 

1.356 

83.6 

0.598 

0.718 

0.837 

0.957 

1.077 

.196 

1.316 

1.435 

*88.6 

0.564 

0.677 

0.790 

0.903 

1.016 

1.129 

1.242 

.354 

83.7 

0.597 

0.717 

0.836 

0.956 

1.075 

.195 

1.314 

1.434 

88.7 

0.564 

0.676 

0.789 

0.902 

1.015 

1.127 

1.240 

.353 

83.8 

0.597 

0.716 

0.835 

0.95-5 

1.074 

.193 

1.313 

1.432 

88.8 

0.563 

0.676 

0.788 

0.901 

1.014 

1.126 

1.239 

.351 

83.9 

0.596 

0.715 

0.834 

0.954 

1.073 

.192 

1.311 

1.430 

88.9 

0.562 

0.675 

0.787 

0.900 

1.012 

1.125 

1.237 

.350 

84.0 

0.595 

0.714 

0.833 

0.952 

1.H71 

1.190 

1.310 

1.429 

89.0 

0.562 

0.674 

0.787 

0.899 

1.011 

1.124 

1.236 

.348 

84.1 

0.595 

0.713 

0.832 

0.951 

1.070 

1.189 

1.308 

1.427 

89.1 

0.561 

ff.673 

0.786 

0.898 

1.010 

1.122 

1.235 

.347 

84.2 

0.594 

0.713 

0.831 

0.950 

I.ii69 

1.188 

1.306 

1.425 

89.2 

0.561 

0.673 

0.785 

0.897 

1.009 

1.121 

1.233 

.345 

84.3 

0.593 

0.712 

0.830 

0.949 

1.;  68 

1.186 

1.305 

1.423 

89.3 

0.560 

0.672 

0.784 

0.896 

1.008 

1.120 

1.232 

1.344 

84.4 

0.592 

0.711 

0.829 

0.948 

1.166 

1.185 

1.303 

1.422 

89.4 

0.559 

0.671 

0.783 

0.895 

1.007 

1.119 

1.230 

1.342 

84.5 

0.592 

0.710 

0.828 

0.947 

1.065 

1.183 

1.302 

1.420 

89.5 

0.559 

0.670 

0.782 

0,894 

1.006 

1.117 

1.229 

1.341 

84.6 

0.591 

0.709 

0.827 

0.946 

1.064 

1.182 

1.300 

1.418 

89.6 

0.558 

0.670 

0.781 

0.893 

1.004 

1.116 

1.228 

1.339 

84.7 

0.590 

0.708 

0.826 

0.945 

1.063 

1.181 

1.299 

1.417 

89.7 

0.557 

0.66  i 

0.780 

0.892 

1.003  1.115 

1.226 

1.338 

84.8 

0.590 

0.708 

0.825 

0.943 

1.061 

1.179 

1.297 

1.415 

89.8 

0.557 

0.66  -i 

0.780 

0.891 

1.002 

1.114   1.225 

1.336 

84.9  0.589 

0.707 

0.824 

0.942 

1.060 

1.178 

1.2961  1.413 

89.9 

0.556 

0.667 

0.779 

0.890 

1.001 

1.112  1.224 

1.335 

240 

TABLE    XXVIII  — CONTINUED. 

TABLE    OP    VELOCITIES    OF    TUBES    IN  MEASURING    FLUMES,   IN    FEET   PER    SECOND.       THE    TIME 

OCCUPIED  IN  PASSING  FROM  THE  UPSTREAM  TO  THE    DOWNSTREAM  TRANSIT 

STATION,  AND  THE   DISTANCE  BETWEEN   THEM,  BEING  GIVEN. 


TIME 
Sec 

DISTANCE  BETWEEN  THE  TRANSIT  STATIONS,  IN  FEET. 

TIME 
See's. 

DISTANCE  BETWEEN  THE  TRANSIT  STATIONS,  IN  FEET. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

50. 

60. 

70. 

80. 

90. 

100. 

110. 

120. 

90.0 

0.556 

0.667 

0.778 

0.889 

1.000 

1.111 

1.222 

1.333 

95.0 

0.5261  0.632 

0.737 

0.842 

!  0.947 

1.053 

1.158 

1.263 

90.1 

0.555 

0.66G 

0.777 

0.888 

0.999 

1.110 

1.221 

1.332 

95.1 

0.526 

0.631 

0.736 

0.841  !  0.946 

1.052i  1.157 

1.262 

90.2 

0.554 

0.665 

0.776 

0.887 

0.998 

1.109 

1.220 

1.330 

95.2 

0.525 

0.630 

0.735 

0.840 

0.945 

1.050 

1.155 

1.261 

90.3 

0.554 

0.664 

0.775 

0.886 

0.997 

1.107 

1.218 

1.329 

95.3 

0.525 

0.630 

0.735 

0.839 

0.944 

1.049 

1.154 

1.259 

90.4 

0.553 

0.664 

0.774 

0.885 

0.996 

1.106 

1.217 

1.327 

95.4 

0.524 

0.629 

0.734 

0.839 

0.943 

1.048 

1.153 

1.258 

90.5 

0.552 

0.663 

0.773 

0.884 

0.994 

1.105 

1.215 

1.326 

95.5 

0.524 

0.628 

0.733 

0.838 

0.942 

1.047 

1.152 

1.257 

90.6 

0.552 

0.662 

0.773 

0.883 

0.993 

1.104 

1.214 

.1.325 

95.6 

0.523 

0.628 

0.732 

0.837 

0.941 

1.046 

1.151 

1.255 

90.7 

0.551 

0.662 

0.772 

0.882 

0.992 

1.103 

1.213 

1.323 

95.7 

0.522 

0.627 

0.731 

0.836 

0.940 

1.045 

1.149 

1.254 

90.8 

0.551 

0.661 

0.771 

0.881 

0.991 

1.101 

1.211 

1.322 

95.8 

0.522 

0.626|  0.731 

0.835 

0.939 

1.044 

1.148 

1.253 

90.9 

0.550 

0.660 

0.770 

0.880 

0.990 

1.100 

1.210 

1.320 

95.9 

0.521 

0.626 

0.730 

0.834 

0.938 

1.043 

1.147 

1.251 

91.0 

0.549 

0.659 

0.769 

0.879 

0.989 

1.099 

1.209 

1.319 

96.0 

0.521 

0.625 

0.729 

0.833 

0.937 

1.042 

1.146 

1.250 

91.1 

0.549 

0.659 

0.768 

0.878 

0.988 

1.098 

1.207 

1.317 

96.1 

0.520 

0.624 

0.728 

0.832 

0.937 

1.041 

1.145 

1.249 

91.2 

0.548 

0.658 

0.768 

0.877 

0.987 

1.096 

1.206 

1.316 

96.2 

0.520 

0.624 

0.728 

0.832 

0.936 

1.040 

1.143 

1.247 

91.3 

0.548 

0.657 

0.767 

0.876 

0.986 

1.095 

1.205 

1.314 

96.3 

0.519 

0.623 

0.727 

0.831 

0.935 

1.038 

1.142 

1.246 

91.4 

0.547 

0.656 

0.766 

0.875 

0.985 

1.094 

1.204 

1.313 

96.4 

0.519 

0.622 

0.726 

0.830 

0.934 

1.037 

1.141 

1.245 

91.5 

0.546 

0.656 

0.765 

0.874 

0.984 

1.093 

1.202 

1.311 

96.5 

0.518 

0.622 

0.725 

0.829 

0.933 

1.036 

1.140 

1.244 

91.6 

0.546 

0.655 

0.764 

0.873 

0.983 

1.092 

1.201 

1.310 

96.6 

0.518 

0.621 

0.725 

0.828 

0.932 

1.035  1.139 

1.242 

91.7 

0.545 

0.654 

0.763 

0.872 

0.981 

1.091 

1.200 

1.309 

96.7 

0.517 

0.620 

0.724 

0.827 

0.931 

1.034 

1.138 

1.241 

91.8 

0.545 

0.654 

0.763 

0.871 

0.980 

1.089 

1.198 

1.307 

96.8 

0.517 

0.620 

0.723 

0.826 

0.930 

1.033 

1.136 

1.240 

91.9 

0.544 

0.653 

0.762 

0.871 

0.979 

1.088 

1.197 

1.306 

96.9 

0.516 

0.619 

0.722 

0.826 

0.929 

1.032 

1.135 

1.238 

92.0 

0.543 

0.652 

0.761 

0.870 

0.978 

1.087 

1.196 

1.304 

97.0 

0.515 

0.619 

0.722 

0.825 

0.928 

1.031 

1.134 

1.237 

92.1 

0.543 

0.651 

0.760 

0.869 

0.977 

1.086 

1.194 

1.303 

97.1 

0.515 

0.618 

0.721 

0.824 

0.927 

1.030 

1.133 

1.236 

92.2 

0.542 

0.651 

0.759 

0.868 

0.976 

1.085 

1.193 

1.302 

97.2 

0.514 

0.617 

0.720 

0.823 

0.926 

1.029 

1.132 

1.235 

92.3 

0.542 

0.650 

0.758 

0.867 

0.975 

1.083 

1.192 

1.300 

97.3 

0.514 

0.617 

0.719 

0.822 

0.925 

1.028 

1.131 

1.233 

92.4 

0.541 

0.649 

0.758 

0.866 

0.974 

1.082 

1.190 

1.299 

97.4 

0.513 

0,616 

0.719 

0.821 

0.924 

1.027 

1.129 

1.232 

92.5 

0.541 

0.649 

0.757 

0.865 

0.973 

1.081 

1.189 

1.297 

97.5 

0.513 

0.615 

0.718 

0.821 

0.923 

1.026 

1.128 

1.231 

92.6 

0.540 

0.648 

0.756 

0.864 

0.972 

1.080 

1.188 

1.296 

97.6 

0.512 

0.615 

0.717 

0.820 

0.922 

1.025 

1.127 

1.230 

92.7 

0.539 

0.647 

0.755 

0.863 

0.971 

1.079 

1.187 

1.294 

97.7 

0.512 

0.614 

0.716 

0.819 

0.921 

1.024 

1.126 

1.228 

92.8 

0.53!) 

0.647 

0.754 

0.862 

0.970 

1.078 

1.185 

1.293 

97.8 

0.511 

0.613 

0.716 

0.818 

0.920 

1.022 

1.125 

1.227 

92.9 

0.538 

0.646 

0.753 

0.861 

0.969 

1.076 

1.184 

1.292 

97.9 

0.511 

0.613 

0.715 

0.817 

0.919 

1.021 

1.124 

1.226 

93.0 

0.538 

0.645 

0.753 

0.860 

0.968 

1.075 

1.183 

1.290 

98.0 

0.510 

0.612 

0.714 

0.816 

0.918 

1.020 

1.122 

1.224 

J3.1 

0.537 

0.644 

0.752 

0.859 

0.967 

1.074 

1.182 

1.289 

98.1 

0.510 

0.612 

0.714 

0.815 

0.917 

1.019 

1.121 

1.223 

93.2 

0.536 

0.644 

0.751 

0.858 

0.966 

1.073 

1.180 

1.288 

98.2 

0.509 

0.611 

0.713 

0.815 

0.916 

1.018 

1.120 

1.222 

93.3 

0.536 

0.643 

0.750 

0.857 

0.965 

1.072 

1.179 

1.286 

98.3 

0.509]  0.610 

0.712 

0.814 

0.916 

1.017 

1.119 

1.221 

93.4 

0.535 

0.642 

0.749 

0.857 

0.964 

1.071 

1.178 

1.285 

98.4 

0.508 

0.610 

0.711 

0.813 

0.915 

1.016 

1.118 

1.220 

93.5 

0.535 

0.642 

0.749 

0.856 

0.963 

1.070 

.176 

1.283 

98.5 

0.508 

0.609 

0.711 

0.812 

0.914 

1.015 

1.117 

1.218 

93.6 

0.534 

0.641 

0.748 

0.855 

0.962 

1.068 

.175 

1.282 

98.6 

0.507 

0.609 

0.710 

0.811 

0.913 

1.014 

1.116 

1.217 

93.7 

0.534 

0.640 

0.747 

0.854 

0.961 

1.067 

.174 

1.281 

98.7 

0.507 

0.608 

0.709 

0.811 

0.912 

1.013 

1.114 

1.216 

93.8 

0.533 

0.640 

0.746 

0.853 

0.959 

1.066 

.173 

1.279 

98.8 

0.506 

0.607 

0.709 

0.810 

0.911 

1.012 

1.113 

1.215 

93.9 

0.532 

0.639 

0.745 

0.852 

0.958 

1.065 

.171 

1.278 

98.9 

0.506 

0.607 

0.708 

0.809 

0.910 

1.011 

1.112 

1.213 

94.0 

0.532 

0.638 

0.745 

0.851 

0.957 

1.064 

1.170 

1.277 

99.0 

0.505 

0.606 

0.707 

0.808 

0.909 

1.010 

1.111 

1.212 

94.1 

0.531 

0.638 

0.744 

0.850 

0.956 

1.063 

1.169 

1.275 

99.1 

0.505 

0.605 

0.706 

0.807 

0.908 

1.009 

1.110 

1.211 

94.2 

0.531 

0.637 

0.743 

0.849 

0.955 

.062 

1.168 

1.274 

99.2 

0.504 

0.605 

0.706 

0.806 

0.907 

1.008 

1.109 

1.210 

94.3 

0.530 

0.636 

0.742 

0.848 

0.954 

.060 

1.166 

1.273 

99.3 

0.504 

0.604 

0.705 

0.806 

0.906 

1.007 

1.108 

1.208 

94.4 

0.530 

0.636 

0.742 

0.847 

0.953 

.059 

1.165 

1.271 

99.4 

0.503 

0.604 

0.704 

0.805 

0.905 

1.006 

1.107 

1.207 

94.5 

0.529 

0.635 

0.741 

0.847 

0.952 

.058 

1.164 

1.270 

99.5 

0.503 

0.603 

0.704 

0.804 

0.905 

1.005 

1.106 

1.206 

94.6 

0.529 

0.634 

0.740 

0.846 

0.951 

.057 

1.163 

1.268 

99.6 

0.502 

0.602 

0.703 

0.803 

0.904 

1.004 

1.104 

1.205 

94.7 

0.528 

0.634 

0.739 

0.845 

0.950 

.056 

1.162 

1.267 

99.7 

0.502 

0.602 

0.702 

0.802 

0.903 

1.003 

1.103 

1.204 

94.8 

0.527 

0.633 

0.738 

0.844 

0.949 

.055 

1.160 

1.266 

99.8 

0.501 

0.601 

0.701 

0.802 

0.902 

1.002 

1.102 

1.202 

94.9 

0.527 

0.632 

0.738 

0.843 

0.948 

1.054 

1.159 

1.264 

99.9 

0.501 

0.601 

0.701 

0.801 

0.901 

1.001 

1.101 

1.201 

100.0 

0.500 

0.600 

0.700 

0.800 

0.900 

1.000  1.100 

1.200 

241 

TABLE    XXIX. 
VALUES  OF  THE  COEFFICIENT  (l  —  0.11G  (y^D  —  0.1)). 


D 

Value  of  the 
Coefficient. 

Logarithm  of  the  Coefficient. 

D 

Value  of  the 
Coefficient. 

Logarithm  of  the  Coefficient. 

0.000 

1.01160 

0.0050088 

0.050 

0.98566 

T.9937271 

0.001 

1.00793 

0.0034304 

0.051 

0.98540 

T.9936126 

0.002 

1.00641 

0.0027749 

0.052 

0.98515 

T.9935024 

0.003 

1.00525 

0.0022741 

0.053 

0.98489 

T.9933877 

0.004 

1.00426 

0.0018462 

0.054 

0.98464 

T.9932775 

0.005 

1.00340 

0.0014741 

0.055 

0.98440 

T.9931716 

0.006 

1.00261 

0.0011320 

0.056 

0.98415 

T.9930613 

0.007 

1.00189 

0.0008200 

0.057 

0.98391 

T.9929554 

0.008 

1.00122 

0.0005295 

0.058 

0.98366 

T.9928450 

0.009 

1.00060 

0.0002605 

0.059 

0.98342 

T.9927390 

0.010 

1.00000 

0.0000000 

0.060 

0.98319 

T.9926375 

0.011 

0.99943 

T.9997524 

0.061 

0.98295 

T.9925314 

0.012 

0.99889 

T.9995177 

0.062 

0.98272 

T.9924298 

0.013 

0.99837 

T.9992915 

0.063 

0.98248 

T.9923237 

0.014 

0.99787 

1.9990740 

0.064 

0.98225 

T.9922220 

0.015 

0.99739 

T.9988650 

0.065 

0.98203 

T.9921248 

0.016 

0.99693 

T.9986647 

0.066 

0.98180 

T.9920230 

0.017 

0.99648 

T.9984686 

0.067 

0.98157 

T.  9919213 

0.018 

0.99604 

T.9982768 

0.068 

0.98135 

T.9918239 

0.019 

0.99561 

T.9980893 

0.069 

0.98113 

T.9917266 

0.020 

0.99520 

T.9979104 

0.070 

0.98091 

T.9916292 

0.021 

0.99479 

T.9977314 

0.071 

0.98069 

1.9915317 

0.022 

0.99439 

T.9975567 

0.072 

0.98047 

T.99  14343 

0.023 

0.99401 

T.9973908 

0.073 

0.98026 

T.9913413 

0.024 

0.99363 

T.9972247 

0.074 

0.98004 

T.9912438 

0.025 

0.99326 

T.9970629 

0.075 

0.97983 

T.9911507 

0.026 

0.99290 

T.9969055 

0.076 

0.97962 

T.9910576 

0.027 

0.99254 

T.9967480 

0.077 

0.97941 

T.9909645 

0.028 

0.99219 

T.9965948 

0.078 

0.97920 

T.9908714 

0.029 

0.99185 

T.9964460 

0.079 

0.97900 

T.9907827 

0.030 

0.99151 

T.9962971 

0.080 

0.97879 

T.9906895 

0.031 

0.99118 

1.9961525 

0.081 

0.97859 

T.9906008 

0.032 

0.99085 

1.9960079 

0.082 

0.97838 

T.9905076 

0.033 

0.99053 

T.9958676 

0.083 

0.97818 

T.9904188 

0.034 

0.99021 

T.9957273. 

0.084 

0.97798 

T.9903300 

0.035 

0.98990 

T.9955913 

0.085 

0.97778 

T.9902411 

0.036 

0.98959 

T.9954553 

0.086 

0.97758 

T.9901523 

0.037 

0.98929 

1.9953236 

0.087 

0.97738 

T.  9900634 

0.038 

0.98899 

T.9951919 

0.088 

0.97719 

T.9899790 

0.039 

0.98869 

T.9950601 

0.089 

0.97699 

1.9898901 

0.040 

0.98840 

T.9949327 

0.090 

0.97680 

1.9898057 

0.041 

0.98811 

T.9948053 

0.091 

0.97661 

T.9897212 

0.042 

0.98783 

T.9946822 

0.092 

0.97641 

1.9896322 

0.043 

0.98755 

1.9945591 

0.093 

0.97622 

1.9895477 

0.044 

0.98727 

T.9944359 

0.094 

0.97604 

1.9894676 

0.045 

0.98699 

T.9943128 

0.095 

0.97585 

1.9893831 

0.046 

0.98672 

T.9941939 

0.096 

0.97566 

1.9892985 

0.047 

0.98645 

T.9940751 

0.097 

0.97547 

1.9892139 

0.048 

0.98619 

T.9939606 

0.098 

0.97529 

1.9891338 

0.049 

0.98592 

T.9938417 

0.099 

0.97510 

1.9890492 

0.050 

0.98566 

T.9937271 

0.100 

0.97492 

1.9889690 

31 


242 


TABLE    XXX. 
TELOCITIES,  IN,  FEET  PER  SECOND,  DUE  TO  HEADS  FROM  0  TO  4.99  FEET. 


Head. 

O 

1 

a 

3 

4 

5 

6 

7 

8 

9 

0.0 

0.000 

0.802 

1.134 

1.389 

1.604 

1.793 

1.965 

2.122 

2.268 

2.406 

.1 

2.536 

2.660 

2.778 

2.892 

3.001 

3.106 

3.208 

3.307 

3.403 

3.496 

.2 

3.587 

3.675 

3.762 

3.846 

3.929 

4.010 

4.090 

4.167 

4.244 

4.319 

.3 

4.393 

4.465 

4.537 

4.607 

4.677 

4.745 

4.812 

4.878 

4.944 

5.009 

.4 

5.072 

5.135 

5.198 

5.259 

5.320 

5.380 

5.440 

5.498 

5.557 

5.614 

.5 

5.G71 

5.728 

5.783 

5.839 

'  5.894 

5.948 

6.002 

6.055 

6.108 

6.100 

.6 

6.212 

6.264 

6.315 

6.366 

6.416 

6.466 

6.516 

6.565 

6.614 

6.662 

.7 

6.710 

6.758 

6.805 

6.852 

6.899 

6.946 

6.992 

7.038 

7.083 

7.129 

.8 

7.173 

7.218 

7.263 

7.307 

7.351 

7.394 

7.438 

7.481 

7.524 

7.566 

.9 

7.609 

7.651 

7.693 

7.734 

7.776 

7.817 

7.858 

7.899 

7.940 

7.980 

1.0 

8.020 

8.060 

8.100 

8.140 

8.179 

8.218 

8.257 

8.296 

8.335 

8.373 

.1 

8.412 

8.450 

8.488 

8.526 

8.563 

8.601 

>     8.638 

8.675 

8.712 

8.749 

.2 

8.78G 

8.822 

8.859 

8.895 

8.931 

8.967 

9.003 

9.038 

9.074 

9.  1  09 

.3 

9.144 

9.180 

9.214 

9.249 

9.284 

9.319 

9.353 

9.387 

9.422 

9.450 

.4 

9.490 

9.523 

9.557 

9.591 

9.624 

9.658 

9.691 

9.724 

9.757 

9.790 

.5 

9.823 

9.855 

9.888 

9.920 

9.953 

9.985 

10.017 

10.049 

10.081 

10.113 

.6 

10.145 

10.176 

10.208 

10.240 

10.271 

10.302 

10.333 

10.364 

10395 

10.426 

.7 

10.457 

10.488 

10.518 

10.549 

10.579 

10.610 

10.640 

10.670 

10.700 

10.730 

.8 

10.760 

10.790 

10.820 

10.850 

10.879 

10.909 

10.938 

10.967 

10.997 

11.026 

9 

11.055 

11.084 

11.113 

11.142 

11.171 

1  1.200 

11.228 

11.257 

11.285 

11.314 

2.0 

11.342 

11.371 

11.399 

1  1  .427 

11.455 

11.483 

11.511 

11.539 

11.567 

11.595 

.1 

11.622 

11.650 

11.678 

11.705 

11.733 

11.760 

11.787 

11.814 

11.842 

11.869 

.2 

11.896 

11.923 

11.950 

11.977 

12.004 

12.030 

12.057 

12.084 

12.110 

12.137 

.3 

12.163 

12.190 

12.216 

12.242 

12.269 

12.295 

12.321 

12.347 

12.373 

12.3'J9 

.4 

12.425 

12.451 

12.477 

12.502 

12.528 

12.554 

12.579 

12.605 

12.630 

12.656 

.5 

12.681 

12.706 

12.732 

12.757 

12.782 

12.807 

12.832 

1  2.857 

12.882 

12.907 

.6 

12.932 

12.957 

12.982 

13.007 

13.031 

13.056 

13.081 

13.105 

13.130 

13.154 

.7 

13.179 

13.203 

13.227 

13.252 

13.276 

13.300 

13.324 

13.348 

13.372 

13.396 

.8 

13.420 

13.444 

13.468 

13.492 

13.516 

13.540 

13.563 

13.587 

13.611 

13.634 

.9 

13.658 

13.681 

13.705 

13.728 

13.752 

13.775 

13.798 

13.822 

13.845 

13868 

3.0 

13.891 

13.915 

13.938 

13.961 

13.984 

14.007 

14.030 

14.053 

14.075 

14.098 

.1 

14.121 

14.144 

14.166 

14.189 

14.212 

14.234 

14.257 

14.280 

14.302 

1  4.325 

.2 

14.347 

14.369 

14.392 

14.414 

14.436 

14.459 

14.481 

14.503 

14.525 

14.547 

.3 

14.569 

14.591 

14.613 

14.635 

14.657 

14.079 

14.701 

14.723 

14.745 

14.767 

.4 

14.789 

14.810 

14.832 

14.854 

14.875 

14.897 

14.918 

14.940 

14.961 

1  4.983 

.5 

15.004 

15.026 

15.047 

15.069 

15.090 

15.111 

15.132 

15.154 

15.175 

15.196 

.6 

15.217 

15.238 

15.J59 

15.281 

15.302 

15.323 

15.344 

15.364 

15.385 

15.406 

.7 

15.427 

15.448 

15.469 

15.490 

15.510 

15.531 

15.552 

15.572 

15.593 

15.614 

.8 

15.634 

15.655 

15.675 

15.696 

15.716 

15.737 

15.757 

15.778 

15.798 

15.818 

.9 

15.839 

15.859 

15.879 

15.899 

15.920 

15.940 

15.960 

15.980 

16.000 

16.020 

4.0 

16.040 

16.060 

16.080 

16.100 

16.120 

16.140 

16.160 

16.180 

16.200 

'16.220 

.1 

16.240 

16.259 

16.279 

1  6.299 

16.319 

1  6.338 

1  6.358 

16.378 

16.397 

16.417 

.2 

16.4:37 

16.456 

16.476 

16.495 

16.515 

16.534 

16.554 

1  6.573 

16.592 

16.612 

.3 

16.631 

16.650 

16.670 

16.689 

16.708 

16.727 

16.747 

16.766 

16.785 

16.804 

.4 

16.823 

16.842 

16.862 

16.881 

16.900 

16.919 

16.938 

16.957 

16.976 

16.994 

.5 

17.013 

17.032 

17.051 

17.070 

17.089 

17.108 

17.126 

17.145 

17.164 

17.183 

.6 

17.201 

17.220 

17.239 

17.257 

17.276 

17.295 

17.313 

17.332 

17.350 

17.369 

.7 

17.387 

17.406 

17.424        17.443 

17.461 

17.480 

17.498 

17.516 

17.535 

17.553 

.8 

17.571 

17.590 

17.608        17.626 

17.644 

17.663 

17.081 

17.699 

17.717 

17.735 

.9 

17.753 

17.772 

17.790       17.808 

17.826 

17.844 

17.862 

17.880 

1  7.898 

17.916 

1 

243 


TABLE    X  X  X  —  CONTINUED. 
VELOCITIES,  IN  FEET  PER  SECOND,  DUE  TO  HEADS  FROM  5  TO   9.99   FEET. 


Head. 

O 

1 

a 

3 

4 

5 

6 

7 

8 

9 

5.0 

17.934 

17.952 

17.970 

17.987 

18.005 

18.023 

18.041 

18.059 

18.077 

18.094 

.1 

18.112 

18.130 

18.148 

18.165 

18.183 

18.201 

18.218 

18.236 

18.254 

18.271 

.2 

18.289 

18.306 

18.324 

18.342 

18.359 

18.377 

18.394 

18.412 

18.429 

18.446 

.3 

18.464 

18.481 

18.499 

18.516 

18.533 

18.551 

18.568 

18.585 

18.603 

18.020 

A 

18.637 

18.655 

18.672 

18.689 

18.706 

18.723 

18.741 

18.758 

18.775 

18.792 

.0 

18.809 

18.826 

18.843 

18.860 

18.877 

18.894 

18.911 

18.928 

18.945 

18.962 

.6 

18.979 

18.996 

19.013 

19.030 

19.047 

19.064 

19.081 

19.098 

19.114 

19.131 

.7 

19.148 

19.165 

19.182 

19.198 

19.215 

19.232 

19.248 

19.265 

19.282 

19.299 

.8 

19.31;) 

19.332 

19.348 

19.365 

19.382 

19.398 

19.415 

19.431 

19.448 

19.464 

.9 

19.481 

19.497 

19.514 

19.530 

19.547 

19.563 

19.580 

19.596 

19.613 

19.629 

6.0 

19.645 

19.662 

19.678 

19.694 

19.711 

19.727 

19.743 

19.760 

19.776 

19.792 

.1 

19.808 

19.825 

19.841 

19.857 

19.873 

19.889 

19.906 

19.922 

19.938 

19.954 

.2 

19.970 

19.986 

20.002 

20.018 

20.034 

20.050 

20.067 

20.083 

20.099 

20115 

.3 

20.131 

20.147 

20.162 

20.178 

20.194 

20.210 

20.226 

20.242 

20.258 

20.274 

A 

20.290 

20.306 

20.321 

20.337 

20.353 

20.369 

20.385 

20.400 

20.416 

20.432 

.5 

20.448 

20.403 

20.479 

20.495 

20.510 

20.526 

20.542 

20.557 

20.573 

20.589 

.6 

20.604 

20.620 

20.635 

20.651 

20.667 

20.682 

20.698 

20.713 

20.729 

20.744 

.7 

20.760 

20.775 

20.791 

20.806 

20.822 

20.837 

20.853 

20.868 

20.883 

20.899 

.8 

20.914 

20.929 

20.945 

20.960 

20.976 

20.991 

21.006 

21.021 

21.037 

21.052 

.9 

21.067 

21.083 

21.098 

21.113 

21.128 

21.144 

21.159 

21.174 

21.189 

21.204 

7.0 

21.219 

21.235 

21.250 

21.265 

21.280 

21.295 

21.310 

21.325 

21.340 

21.355 

.1 

21.370 

21.386 

21.401 

21.416 

21.431 

21.446 

21.461 

21.476 

21.491 

21.500 

.2 

21.520 

21.535 

21.550 

21.565 

21.580 

21.595 

21.610 

21.625 

21.640 

21.655 

.3 

21.669 

21.684 

21.699 

21.714 

21.729 

21.743 

21.758 

21.773 

21.788 

21.803 

A 

21.817 

21.832 

21.847 

21.861 

21.876 

21.891 

21.906 

21.920 

21.935 

2  1  .950 

.5 

21.964 

21.979 

21.993 

22.008 

22.023 

22.037 

22.052 

"22.066 

22.081 

22.096 

.6 

22.110 

22.125 

22.139 

22.154 

22.168 

22.183 

22.197 

22.212 

22.226 

22.241 

.7 

22.255 

22.270 

22.284 

22.298 

22.313 

22.327 

22.342 

22.356 

22.370 

22.385 

.8 

22.399 

22.414 

22.428 

22.442 

22.457 

22.471 

22.485 

22.499 

22.514 

22.528 

.9 

22.542 

22.557 

22.571 

22.585 

22.599 

22.614 

22.628 

22.642 

22.656 

22.670 

8.0 

22.685 

22.699 

22.713 

22.727 

22.741 

22.755 

22.769 

22.784 

22.798 

22.812 

.1 

22.826 

22.840 

22.854 

22.868 

22.882 

22.896 

22.910 

22.924 

22.938 

22.952 

.2 

22.966 

22.980 

22.994 

23.008 

23.022 

23.036 

23.050 

23.064 

23.078 

23.092 

.3 

23.106 

23.120 

23.134 

23.148 

23.162 

23.175 

23.189 

23.203 

23.217 

23.231 

.4 

23.245 

23.259 

23.272 

23.286 

23.300 

23.314 

23.328 

23.341 

23.355 

23.369 

.5 

23.383 

23.396 

23.410 

23.424 

23.438 

23.451 

23.405 

23.479 

23.492 

23.506 

.6 

23.520 

23.534 

23.547 

23.561 

23.574 

23.588 

23.602 

23.615 

23.629 

23.643 

.7 

23.656 

23.670 

23.683 

23.697 

23.711 

23.724 

23.738 

23.751 

23.765 

23.778 

.8 

23.792 

23.805 

23.819 

23.832 

23.846 

23.859 

23.873 

23.886 

23.900 

23.913 

.9 

23.927 

23.940 

23.953 

23.967 

23.980 

23.994 

24.007 

24.020 

24.034 

24.047 

9.0 

24.061 

24.074 

24.087 

24.101 

24.114 

24.127 

24.141 

24.154 

24.167 

24.181 

.1 

24.194 

24.207 

2-4.220 

24.234 

24.247 

24.260 

24.274 

24.287 

24.300 

24.313 

.2 

24.326 

24.340 

24.353 

24.366 

24.379 

24.392 

24.406 

24.419 

24.432 

24.445 

.3 

24.458 

24.471 

24.485 

24.498 

24.511 

24.524 

24.537 

24.550 

24.563 

24.576 

.4 

24.589 

24.603 

24.616 

24.629 

24.642 

24.655 

24.668 

24.681 

24.694 

24.707 

,5 

24.720 

24.733 

24.746 

24.759 

24,772 

24.785 

24.798 

24.811 

24.824 

24.837 

.6 

24.850 

24.803 

24.876 

24.888 

24.901 

24.914 

24.927 

24.940 

24.953 

24.966 

.7 

24.979 

24.992 

25.005 

25.017 

25.030 

25.043 

25.056 

25.069 

25.082 

25.094 

.8 

25.107 

25.120 

25.133 

25.146 

25.158 

25.171 

25.184 

25.197 

25.209 

25222 

.9 

25.235 

25.248 

25.260 

25.273 

25.286 

25.299 

25.311 

25.324 

25.337 

25.349 

244 


TABLE    XXX— CONTINUED. 
VELOCITIES,  IN  FEET  PER  SECOND,  DUE  TO   HEADS  FROM  10  TO  14.99  FEET. 


Ilead. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10.0 

25.362 

25.375 

25.387 

25.400 

25.413 

25.425 

25.438 

25.451 

25.463 

25.476 

.1 

25.489 

25.501 

25.514 

25.526 

25.539 

25.552 

25.564 

25.577 

25.589 

25.602 

.2 

25.614 

25.627 

25.640 

25.652 

25.665 

25.677 

25.690 

25.702 

25.715 

25.728 

.3 

25.740 

25.752 

25.765 

25.777 

25.790 

25.802 

25.815 

25.827 

25.839 

25.852 

.4 

25.864 

25.877 

25.889 

25.902 

25.914 

25.926 

25.939 

25.951 

25.964 

25.976 

.5 

25.988 

26.001 

26.013 

26.026 

26.038 

26.050 

26.063 

26.075 

26.087 

26.099 

.6 

26.112 

26.124 

26.136 

26.149 

26.161 

26.173 

26.186 

26.198 

26.210 

26.222 

.7 

26.235 

26.247 

26.259 

26.272 

.26.284 

26.296 

26.308 

26.320 

26.333 

26.345 

.8 

20.357 

26.369 

26.381 

26.394 

26.406 

26.418 

26.430 

26.442 

26.454 

26.467 

.9 

26.479 

26.491 

26.503 

26.515 

26.527 

26.540 

26.552 

26.564 

26.576 

26.588 

11.0 

26.600 

26.612 

26.624 

26.636 

26.648 

26.660 

26.672 

26.684 

26.697 

26.709 

.1 

26.721 

26.733 

26.745 

26.757 

26.769 

26.781 

26.793 

26.805 

26.817 

26.829 

.2 

26.841 

26.853 

26.865 

26.877 

26.889 

26.901 

26.913 

26.924 

26.936 

26.948 

.3 

26.960 

26.972 

26.984 

26.996 

27.008 

27.020 

27.032 

27.044 

27.056 

27.067 

.4 

27.079 

27.091 

27.103 

27.115 

27.127 

27.139 

27.150 

27.162 

27.174 

27.186 

.5 

27.198 

27.210 

27.221 

27.233 

27.245 

27.257 

27.269 

27.280 

27.292 

27.304 

.6 

27.316 

27.328 

27.339 

27.351 

27.363 

27.375 

27.386 

27.398 

27.410 

27.422 

.7 

27.433 

27.445 

27.457 

27.468 

27.480 

27.492 

27.504 

27.515 

27.527 

27.539 

.8 

27.550 

27.562 

27.574 

27.585 

27.597 

27.609 

27.620 

27.632 

27.644 

27.655 

9 

27.667 

27.678 

27.690 

27.702 

27.713 

27.725 

27.736 

27.748 

27.760 

27.771 

12.0 

27.783 

27.794 

27.806 

27.817 

27.829 

27.841 

27.852 

27.864 

27.875 

27.887 

.1 

27.898 

27.910 

27.921 

27.933 

27.944 

27.956 

27.967 

27.979 

27.990 

28.002 

.2 

28.013 

28.025 

28.036 

28.048 

28.059 

28.071 

28.082 

28.094 

28.105 

28.117 

.3 

28.128 

28.139 

28.151 

28.162 

28.174 

28.185 

28.196 

28.208 

28.219 

28.231 

.4 

28.242 

28.253 

28.265 

28.276 

28.288 

28.299 

28.310 

28.322 

28.333 

28.344 

.5 

28.356 

28.367 

28.378 

28.390 

28.401 

28.412 

28.424 

28.435 

28.446 

28.458 

.6 

28.469 

28.480 

28.491 

28.503 

28.514 

28.525 

28.537 

'28.548 

28.559 

28.570 

.7 

28.582 

28.593 

28.604 

28.615 

28.627 

28.638 

28.649 

28.660 

28.672 

28.683 

.8 

28.694 

28.705 

28.716 

28.727 

28.739 

28.750 

28.761 

28.772 

28.783 

28.795 

.9 

28.806 

28.817 

28.828 

28.839 

28.850 

28.862 

28.873 

28.884 

28.895 

28.906 

13.0 

28.917 

28.928 

28.939 

28.951 

28.962 

28.973 

28.984 

28.995 

29.006 

29.017 

.1 

29.028 

29.039 

29.050 

29.061 

29.073 

29.084 

29.095 

29.106 

29.117 

29.128 

.2 

29.139 

29.150 

29.161 

29.172 

29.183 

29.194 

29.205 

29.2  1  6 

29.227 

29.238 

.3 

29.249 

29.260 

29.271 

29.282 

29.293 

29.304 

29.315 

29.326 

29.337 

29.348 

.4 

29.359 

29.370 

29.381 

29.392 

29.403 

29.413 

29.424 

29.435 

29.446 

29.457 

.5 

29.468 

29.479 

29.490 

29.501 

29.512 

29.523 

29.533 

29.544 

29.555 

29.566 

.6 

29.577 

29.588 

29.599 

29.610 

29.620 

29.631 

29.642 

29.653 

29.664 

29.675 

.7 

29.686 

29.696 

29.707 

29.718 

29.729 

29.740 

29.751 

29.761 

29.772 

29.783 

.8 

29.794 

29.805 

29.815 

29.826 

29.837 

29.848 

29.858 

29.869 

29.880 

29.891 

.9 

29.901 

29.912 

29.923 

29.934 

29.944 

29.955 

29.966 

29.977 

29.987 

29.998 

14.0 

30.009 

30.020 

30.030 

30.041 

30.052 

30.062 

30.073 

30.084 

30.094 

30.105 

.1 

30.116 

30.126 

30.137 

30.148 

30.159 

30.169 

30.180 

30.190 

30.201 

30.212 

.2 

30.222 

30.233 

30.244 

30.254 

30.265 

30.276 

30.286 

30.297 

30.307 

30.318 

.3 

30.329 

30.339 

30.350 

30.360 

30.371 

30.382 

30.392 

30.403 

30.413 

30.424 

.4 

30.435 

30.445 

30.456 

30.466 

30.477 

30.487 

30.498 

30.508 

30.519 

30.529 

.5 

30.540 

30.551 

30.561 

30.572 

30.582 

30.593 

30.603 

30.614 

30.624 

30.635 

.6 

30.64a 

30.656 

30.666 

30.677 

30.687 

30.698 

30.708 

30.719 

30.729 

30.739 

.7 

30.750 

30.760 

30.771 

30.781 

30.792 

30.802 

30.813 

30.823 

30.833 

30.844 

.8 

30.854 

30.865 

30.875 

30.886 

30.896 

30.906 

30.917 

30.927 

30.938 

30.948 

.9 

30.958 

30.969 

30.979 

30,990 

31.000 

31.010 

31.021 

31.031 

31.041 

31.052 

245 


TABLE    X  X  X  —  CONTINUED. 
VELOCITIES,  IN  FEET  PER  SECOND,  DUE  TO  HEADS  FROM  15   TO   19.99  FEET. 


Head. 

O 

1 

3 

3 

4 

5 

6 

7 

8 

9 

15.0 

31.062 

31.072 

31.083 

31.093 

31.103 

31.114 

31.124 

31.134 

31.145 

31.155 

.1 

31.165 

31.176 

31.186 

31.196 

31.207 

31.217 

31.227 

31.238 

31.248 

31.258 

.2 

31.268 

31.279 

31.289 

31.299 

31.310 

31.320 

31.330 

31.340 

31.351 

31.361 

.3 

31.371 

31.381 

31.392 

31.402 

31.412 

31.422 

31.433 

31.443 

31.453 

31.463 

.4 

31.474 

31.484 

31.494 

31.504 

31.514 

31.525 

31.535 

31.545 

31.555 

31.565 

.5 

31.576 

31.586 

31.596 

31.606 

31.616 

31.626 

31.637 

31.647 

31.657 

31.667 

.6 

31.677 

31.687 

31.698 

31.708 

31.718 

31.728 

31.738 

31.748 

31.758 

31.768 

.7 

31.779 

31.789 

31.799 

31.809 

31.819 

31.829 

31.839 

31.849 

31.859 

31.870 

.8 

31.880 

31.890 

31.900 

31.910 

31.920 

31.930 

31.940 

31.950 

31.960 

31.970 

.9 

31.980 

31.990 

32.000 

32.011 

32.021 

32.031 

32.041 

32.051 

32.061 

32.071 

16.0 

32.081 

32.091 

32.101 

32.111 

32.121 

32.131 

32.141 

32.151 

32.161 

32.171 

.1 

32.181 

32.191 

32.201 

32.211 

32.221 

32.231 

32.241 

32.251 

32.261 

32271 

.2 

32.281 

32.291 

32.301 

32.311 

32.321 

32.330 

32.340 

32.350 

32.360 

32.370 

.3 

32.380 

32.390 

32.400 

32.410 

32.420 

32.430 

32.440 

32.450 

32.460 

32.470 

.4 

32.480 

32.489 

32.499 

32.509 

32.519 

32.529 

32.539 

32.549 

32.559 

32.569 

.5 

32.579 

32.588 

32.598 

32.608 

32.618 

32.628 

32.637 

32.647 

32.657 

32.667 

.6 

32.677 

32.687 

32.696 

32.706 

32.716 

32.726 

32.736 

32.746 

32.755 

32.765 

.7 

32.775 

32.785 

32.795 

32.804 

32.814 

32.824 

32.834 

32.844 

32.854 

32.863 

.8 

32.873 

32.883 

32.893 

32.903 

32.912 

32.922 

32.932 

32.941 

32.951 

32.961 

.9 

32.971 

32.980 

32.990 

33.000 

33.010 

33.019 

33.029 

33.039 

33.049 

33.058 

17.0 

33.068 

33.078 

33.088 

33.097 

33.107 

33.117 

33.126 

33.136 

33.146 

33.156 

.1 

33.165 

33.175 

33.185 

33.194 

33.204 

33.214 

33.223 

33.233 

33.243 

33.252 

.2 

33.262 

33.272 

33.281 

33.291 

33.301 

33.310 

33.320 

33.330 

33.339 

33.349 

.3 

33.359 

33.368 

33.378 

33.388 

33.397 

33.407 

33.416 

33.426 

33.436 

33.445 

.4 

33.455 

33.465 

33.474 

33.484 

33.493 

33.503 

33.513 

33.522 

33.532 

33.541 

.5 

33.551 

33.560 

33.570 

33.580 

33.589 

33.599 

33.608 

33.618 

33.628 

33.637 

.6 

33.647 

33.656 

33.666 

33.675 

33.685 

33.694 

33.704 

33.713 

33.723 

33.733 

.7 

33.742 

33.752 

33.761 

33.771 

33.780 

33.790 

33.799 

33.809 

33.818 

33.828 

.8 

33.837 

33.847 

33.856 

33.866 

33.875 

33.885 

33.894 

33.904 

33.913 

33.923 

.9 

33!932 

33.942 

33.951 

33.961 

33.970 

33.980 

33.989 

33.998 

34.008 

34.017 

18.0 

34.027 

34.036 

34.046 

34.055 

34.065 

34.074 

34.083 

34.093 

34.102 

34.112 

.1 

34.121 

34.131 

34.140 

34.149 

34.159 

34.168 

34.178 

34.187 

34.197 

34.206 

.2 

34.215 

34.225 

34.234 

34.244 

34.253 

34.262 

34.272 

34.281 

34.290 

34.300 

.3 

34.309 

34.319 

34.328 

34.337 

34.347 

34.356 

34.365 

34.375 

34.384 

34.393 

.4 

34.403 

34.412 

34.422 

34.431 

34.440 

34.450 

34.459 

34.468 

34.478 

34.487 

.5 

34.496 

34.505 

34.515 

34.524 

34.533 

34.543 

34.552 

34.561 

34.571 

34.580 

.6 

34.589 

34.599 

34.608 

34.617 

34.626 

34.636 

34.645 

34.654 

34.664 

34.673 

.7 

34.682 

34.691 

34.701 

34.710 

34.719 

34.728 

34.738 

34.747 

34.756 

34.766 

.8 

34.775 

34.784 

34.793 

34.802 

34.812 

34.821 

34.830 

34.839 

34.849 

34.858 

.9 

34.867 

34.876 

34.886 

34.895 

34.904 

34.913 

34.922 

34.932 

34.941 

34.950 

19.0 

34.959 

34.968 

34.978 

34.987 

34.996 

35.005 

35.014 

35.024 

35.033 

35.042 

.1 

35.051 

35.0GO 

35.069 

85.079 

35.088 

35.097 

35.106 

35.115 

35.124   35.134 

.2 

35.143 

35.152 

35.161 

35.170 

85.179 

35.188 

35.198 

35.207 

35.216 

35.225 

.3 

35.234 

35.243 

35.252 

35.262 

35.271 

35.280 

35.289 

35.298 

35.307 

35.316 

.4 

35.325 

35.334 

35.344 

35.353 

35.362 

35.371 

35.380 

35.389 

35.398 

35.407 

.5 

35.416 

35.425 

35.434 

35.443 

35.453 

35.462 

35.471 

35.480 

35.489 

35.498 

.6 

35.507 

35.516 

35.525 

35.534 

35.543 

35.552 

35.561 

35.570 

35.579 

35.588 

.7 

35.597 

35.606 

35.615 

35.624 

35.634 

35.643 

35.652 

35.661 

35.670 

35.679 

.8 

35.688 

35.697 

35.706 

35.715 

35.724 

35.733 

35.742 

35.751 

35.760 

35.769 

.9 

35.778 

35.787 

35.796 

35.805 

35.814 

35.823 

35.832 

35.841 

35.849 

35.858 

TABLE    XXX— CONTINUED. 
VELOCITIES,  IN  FEET  PER  SECOND,  DUE  TO   HEADS  FROM  20  TO  24.99  FEET. 


Head. 

0 

1                2 

| 

3 

4 

5 

6 

7 

8 

9 

20.0 

35.8C7 

35.876 

35.885 

35.894 

35.903 

35.912 

35.921 

35.930 

35.939 

35.948 

.1 

35.957 

35.966 

35.975 

35.9<U 

35.993 

36.002 

36.011 

36.020 

36.028 

36.037 

.2 

36.046 

36055 

36.064 

36.073 

36.082 

36.091 

36.100 

36.109 

36.118 

36.127 

.3 

36.135 

36.144 

36.153 

36.162 

36.171 

36.180 

36.189 

36.198 

36.207 

36.215 

.4 

36.224 

36.233 

36.242 

36.251 

36.260 

36.269 

36.278 

36.286 

36.295 

36.304 

.5 

36.313 

36.322 

36.331 

36.340 

36.348 

36.357 

36.366 

36.375 

36.384 

36.393 

.6 

36.401 

36410 

36.419 

36.428 

36.437 

36.446 

36.454 

36.463 

36.472 

36.481 

.7 

30.490 

36.499 

36.507 

36.516 

36.525 

36.534 

36.543 

36551 

36.560 

36.569 

.8 

36578 

36.587 

36.595 

36.604 

36.613 

36.622 

36.630 

36.639 

36.648 

36.657 

.9 

36.666 

36.674 

36.683 

36.692 

36.701 

36.709 

36.718 

36.727 

36.736 

36.744 

21.0 

36.753 

36.762 

36.771 

36.779 

36.788 

36.797 

36.806 

36.814 

36.82.5 

36.832 

.1 

36.841 

36.849 

36.858 

36.867 

36.875 

36.884 

36.893 

36.902 

36.910 

36.919 

.2 

36.928 

36.936 

36.945 

36.954 

36.963 

36971 

36.980 

36.989 

36.997 

37.006 

.3 

37.015 

37.023 

37.032 

37.041 

37.049 

37.058 

37.067 

37.076 

37.084 

37.093 

.4 

37.102 

37.110 

37.119 

37.128 

37.136 

37.145 

37.154 

37.162 

37.171 

37.179 

.5 

37.188 

•37.197 

37.205 

37.214 

37.223 

37.231 

37.240 

37.249 

37.257 

37.266 

.6 

37.275 

37.283  1     37.292 

37.300 

37.309 

37.318 

37.326 

37.335 

37.343 

37.352 

.7 

37.361 

37.369 

37.378 

37.387 

37.395 

37.404 

37.412 

37.421 

37.430 

37.438 

.8 

37.447 

37.455 

37.464 

37.472 

37.481 

37.490 

37.498 

37.506 

37.515 

37.524 

9 

37.532 

37.541 

37.550 

37.558 

37.567 

37.575 

37.581 

37.592 

37.601 

37.610 

22.0 

37.618 

37.627 

37.635 

37.644 

37.652 

37.661 

37.669 

37.678 

37.686 

37.695 

.1 

37.703 

37.712 

37.721 

37.729 

37.738 

37.746 

37.755 

37.763 

37.772 

37.780 

.2 

37.789 

37.797 

37.806 

37.814 

37.823 

37.832 

37.840 

37.848 

37.857 

37.865 

.     -3 

37.874 

37.882 

37.891 

37.899 

37.908 

37.916 

37.925 

37.933 

37.942 

37.950 

.4 

37.959 

37.967 

37.975 

37.984 

37.992 

38.001 

38.009 

38.018 

38.026 

38.035 

.5 

38.043 

38.052 

38.060 

38.068 

38.077 

38.085 

38.094 

38.102 

38.111 

38.119 

.6 

38.128 

38.136 

38.144 

38.153 

38.161 

38.170 

38.178 

38.187 

38.195 

38.203 

.7 

38.212 

38.220 

38.229 

38.237 

38.246 

38.254 

38.262 

38.271 

38.279 

38.288 

.8 

38.296 

38.304 

38.313 

38.321 

38.330 

38.3:38 

38.346 

38.355 

38.363 

38.371 

.9 

38.380 

38388 

38.397 

38.405 

38.413 

38.422 

38.430 

38.438 

38.447 

38.455 

23.0 

38.464 

38.472 

38.480 

38.489 

38.497 

38.505 

38.514 

38.522 

38.530 

38.539 

.1 

38.547 

38.555 

38.564 

38.572 

38.580 

38.589 

38.597 

38.605 

38.614 

38.622 

.2 

38.630 

38.638 

38.647 

38.655 

38.664 

38.672 

38.680 

38.689 

38.697 

38.705 

.3 

38.714 

38.722 

38.730 

38.738 

38.747 

38.755 

38.763 

38.772 

38.780 

38.788 

.4 

38.797 

38.805 

38.813 

38.821 

38.830 

38.838 

38.846 

38.855 

38.863 

38.871 

.5 

38.879 

38.888 

38.896 

38.904 

38.912 

38.921 

38.929 

38.937 

,  38.945 

38.954 

.6 

38.962 

38.970 

38.978 

38.987 

38.995 

39.003 

39.011 

39.020 

39.028 

39.036 

.7 

39.044 

39.053 

39.061 

39.069 

39.077 

39.086 

39.094 

39.102 

39.110 

39.1  i9 

.8 

39.127 

39.135 

39.143 

39.151 

39.160 

39.168 

39.176 

39.184 

39.192 

39.201 

.9 

39.209 

39.217 

39.225 

39.233 

39.242 

39.250 

39.258 

39.266 

39.274 

39.283 

24.0 

39.291 

39.299 

39.307 

39.315 

39.324 

39.332 

39.340 

39.348 

39.356 

39.364 

.1 

39.373 

39.381 

39.389 

39.397 

39.405 

39.413 

39.422 

39.430 

39.438 

39.446 

.2 

39.454 

39.462 

39.470 

39.479 

39.487 

39.495 

39.503 

39.511 

39.519 

39.527 

.3 

39.536 

39.544 

39.552 

39.560 

39.568 

39.576 

39.584 

39.592 

39.601 

39.609 

.4 

39.617 

39.625 

39.633 

39.641 

39.649 

39.657 

39.666 

39.674 

39.682 

39.690 

.5 

39.698 

39.706 

39.714 

39.722 

39.730 

39.738 

39.747 

39.755 

39.763 

39.771 

.6 

39.779 

39.787 

39.795 

39.803 

39.811 

39.819 

39.827 

39.835 

39.844 

39.852 

.7 

39.860 

39.868 

39.876 

39.884 

39.892 

39.900 

39.908 

39.916 

39.924 

39.932 

.8 

39.940 

39.948 

39.956 

39.964 

39.972 

39.981 

39.989 

39.997 

40.005 

40.013 

.9 

40.021 

40.029 

40.037 

40.045 

40.053 

40.061 

40.069 

40.077 

40.085 

40.093 

247 


TABLE    X  X  X  —  CONTINUED. 
VELOCITIES,  IN  FEET  PER  SECOND,  DUE  TO  HEADS  FROM  25  TO  29.99  FEET. 


Head. 

O 

1 

a 

3 

4 

5 

O 

7 

8 

9 

25.0 

40.101 

40.109 

40.117 

40.125 

40.133 

40.141 

40.149 

40.157 

40.165 

40.173 

.1 

40.181 

40.189 

40.197 

40.205 

40.213 

40.221 

40.229 

40.237 

40.245 

40.253 

.2 

40.261 

40.269 

40.277 

40.285 

40.293 

40.301 

40.309 

40.317 

40.325 

40.333 

.3 

40.341 

40.349 

40.357 

40.365 

40.373 

40.381 

40.389 

40.397 

40.405 

40.413 

.4 

40.421 

40.428 

40.436 

40.444 

40.452 

40.460 

40.468 

40.476 

40.484 

40.492 

.5 

40.500 

40.508 

40.516 

40.524 

40.532 

40.540 

40.548 

40.556 

40.563 

40.571 

.6 

40.579 

40.587 

40.595 

40.603 

40.611 

40.619 

40.627 

40.635 

40613 

40.651 

.7 

40.659 

40.666 

40.674 

40.682 

40.690 

40.698 

40.706 

40.714 

40.722 

40.730 

.8 

40.738 

40.745 

40.753 

40.761 

40.769 

40.777 

40.785 

40.793 

40.801 

40.809 

.9 

40.816 

40.824 

40.832 

40.840 

40.848 

40.856 

40.864 

40.872 

40.879 

40.887 

26.0 

40.895 

40.903 

40.911 

40.919 

40.927 

40.934 

40.942 

40.950 

40.958 

40.966 

.1 

40.974 

40.982 

40.989 

40.997 

41.005 

41.013 

41.021 

41.029 

41.036 

41.044 

.2 

41.052 

41.060 

41.068 

41.076 

41.083 

41.091 

41.099 

41.107 

41.115 

41.123 

.3 

41.130 

41.138 

41.146 

41.154 

41.162 

41.169 

41.177 

41.185 

41.193 

41.201 

.4 

41.209 

41.216 

41.224 

41.232 

41.240 

41.248 

41.255 

41.263 

41.271 

41.279 

.5 

41.287 

41.294 

41.302 

41.310 

41.318 

41.325 

41.333 

41.341 

41.349 

41.357 

.6 

41.364 

41.372 

41.380 

41.388 

41.395 

41.403 

41.411 

41.419 

41.426 

41.434 

.7 

41.442 

41.450 

41.458 

41.465 

41.473 

41.481 

41.489 

41.496 

41.504 

41.512 

.8 

41.520 

41.527 

41.535 

41.543 

41.551 

41.558 

41.566 

41.574 

41.581 

41.589 

.9 

41.597 

41.605 

41.612 

41.620 

41.628 

41.636 

41.643 

41.651 

41.659 

41.666 

27.0 

41.674 

41.682 

41.690 

41.697 

41.705 

41.713 

41.720 

41.728 

41.736 

41.744 

.1 

41.751 

41.759 

41.767 

41.774 

41.782 

41.790 

41.797 

41.805 

41.813 

41.821 

.2 

41.828 

41.836 

41.844 

41.851 

41.859 

41.867 

41.874 

41.882 

41.890 

41.897 

.3 

41.905 

41.913 

41.920 

41.928 

41.936 

41.943 

41.951 

41.959 

41.967 

41.974 

.4 

41.982 

41.989 

41.997 

42.005 

42.012 

42.020 

42.028 

42.035 

42.043 

42.051 

.5 

42.058 

42.066 

42.074 

42.081 

42.089 

42.096 

42.104 

42.112 

42.119 

42.127 

.6 

42.135 

42.142 

42.150 

42.158 

42.165 

42.173 

42.180 

42.188 

42.196 

42.203 

.7 

42.211 

42.219 

42.226 

42.234 

42.241 

42.249 

42.257 

42.264 

42.272 

42.279 

.8 

42.287 

42.295 

42.302 

42.310 

42.317 

42.325 

42.333 

42.340 

42.318 

42.355 

.9 

42.363 

42.371 

42.378 

42.386 

42.393 

42.401 

42.409 

42.416 

42.424 

42.431 

28.0 

42.439 

42.446 

42.454 

42.462 

42.469 

42.477 

42.484 

42.492 

42.499 

42.507 

.1 

42.515 

42.522 

42.530 

42.537 

42.545 

42.552 

42.560 

42.568 

42.575 

42.583 

.2 

42.590 

42.598 

42.605 

42.613 

42.620 

42.628 

42.635 

42.643 

42.651 

42.658 

.3 

42.666 

42.673 

42.681 

42.688 

42.696 

42.703 

42.711 

42.718 

42.726 

42.733 

.4 

42.741 

42.748 

42.756 

42.764 

42.771 

42.779 

42.786 

42.794 

42.80  1 

42.809 

.5 

42.816 

42.824 

42.831 

42.839 

42.846 

42.854 

42.861 

42.869 

42.876 

42.884 

.6 

42.891 

42.899 

42.906 

42.914 

42.921 

42.929 

42.936 

42.944 

42.95  1 

42.959 

.7 

42.966 

42.974 

42.981 

42.989 

42.996 

43.004 

43.011 

43.018 

43.026 

43.033 

.8 

43.041 

43.048 

43.056 

43.063 

43.071 

43.078 

43.086 

43.093 

43.101 

43.108 

.9 

43.116 

43.123 

43.130 

43.138 

43.145 

43.153 

43.160 

43.168 

43.175 

43.183 

29.0 

43.190 

43.198 

43.205 

43.212 

43.220 

43.227 

43.235 

43.243 

43.250 

43.257 

.1 

43.264 

43.272 

43.279 

43.287 

43.294 

43.302 

43.309 

43.316 

43.324 

43.331 

.2 

43.339 

43.346 

43.354 

43.361 

43.368 

43.376 

43.383 

43.391 

43.398 

43.405 

.3 

43.413 

43.420 

43.428 

43.435 

43.443 

43.450 

43.457 

43.465 

43.472 

43.480 

.4 

43.487 

43.494 

43.502 

43.509 

43.517 

43.524 

43.531 

43.539 

43.546 

43.553 

.0 

43.561 

43.568' 

43.576 

43.583 

43.590 

43.598 

43.605 

43.612 

43.620 

43.627 

.6 

43.635 

43.642 

43.649 

43.657 

43.664 

43.671 

43.679 

43.686 

43.69  4 

43.701 

.7 

43.708 

43.716 

43.723 

43.730 

43.738 

43.745 

43.752 

43.760 

43.767 

43.774 

.8 

43.782 

43.789 

43.796 

43.804 

43.811 

43.818 

43.826 

43.833 

43.840 

43.848 

.9 

43.855 

43.862 

43.870 

43.877 

43.884 

43.892 

43.899 

43.906 

43.914 

43.921 

I 

248 


TABLE    XXX  —  CONTINUED. 
VELOCITIES,  IN  FEET  PER  SECOND,  DUE  TO  HEADS  FROM  30  TO  34.99  FEET. 


Head. 

O 

1 

9 

3 

4 

5 

G 

7 

8 

r 
9 

30.0 

43.928 

43.936 

43.943 

43.950 

43.958 

43.965 

43.972 

43.980 

43.987 

43.994 

.1 

44.002 

44.009 

44.016 

44.024 

44.031 

44.038 

44.045 

44.053 

44.060 

44.067 

.2 

44.075 

44.082 

44.089 

44.097 

44.104 

44.111 

44.118 

44.126 

44.133 

44.140 

.3 

44.148 

44.155 

44.162 

44.169 

44.177 

44.184 

44.191 

44.198 

44.206 

44.213 

.4 

44.220 

44.228 

44.235 

44.242 

44.249 

44.257 

44.264 

44.271 

44.278 

44.286 

.5 

44.293 

44.300 

44.308 

44.315 

44.322 

44.329 

44.337 

44.344 

44.351 

44.358 

.6 

44.366 

44.373 

44.380 

44.387 

44.395 

44.402 

44.409 

44.416 

44.423 

44.431 

.7 

44.438 

44.445 

44.452 

44.460 

44.46,7 

44.474 

44.481 

44.489 

44.496 

44.503 

.8 

44.510 

44.518 

44.525 

44.532 

44.539 

44.546 

44.554 

44.561 

44.568 

44.575 

.9 

44.582 

44.590 

44.597 

44.604 

44.611 

44.619 

44.626 

44.633 

44.640 

44.647 

31.0 

44.655 

44.662 

44.669 

44.676 

44.683 

44.691 

44.698 

44.705 

44.712 

44.719 

.1 

44.727 

44.734 

44.741 

44.748 

44.755 

44.762 

44.770 

44.777 

44784 

44.791 

.2 

44.798 

44.806 

44.813 

44.820 

44.827 

44.834 

44.841 

44.849 

44.856 

44.863 

.3 

44.870 

44.877 

44.884 

44.892 

44.899 

44.906 

44.913 

44.920 

44.927 

44.935 

.4 

44.942 

44.949 

44.956 

44.963 

44.970 

44.978 

44.985 

44.992 

44.999 

45.006 

.5 

45.013 

45.020 

45.028 

45.035 

45.042 

45.049 

45.056 

45.063 

45.070 

45.078 

.6 

45.085 

45.092 

45.099 

45.106 

45.113 

45.120 

45.127 

45.135 

45.142 

45.149 

.7 

45.156 

45.163 

45.170 

45.177 

45.184 

45.192 

45.199 

45.206 

45.213 

45.220 

.8 

45.227 

45.234 

45.241 

45.248 

45.256 

45.263 

45.270 

45.277 

45.284 

45.291 

.9 

45.298 

45.305 

45.312 

45.319 

45.327 

45.334 

45.341 

45.348 

45.355 

45.362 

32.0 

45.369 

45.376 

45.383 

45.390 

45.397 

45.405 

45.412 

45.419 

45.426 

45.433 

.1 

45.440 

45.447 

45.454 

45.461 

45.468 

45.475 

45.482 

45.489 

45.497 

45.504 

.2 

45.511 

45.518 

45.525 

45.532 

45.539 

45.546 

45.553 

45.560 

45.567 

45.574 

.3 

45.581 

45.588 

45.595 

45.602 

45.609 

45.617 

45.624 

45.631 

45.638 

45.645 

.4 

45.652 

45.659 

45.666 

45.673 

45.680 

45.687 

45.694 

45.701 

45.708 

45.715 

.5 

45.722 

45.729 

45.736 

45.743 

45.750 

45.757 

45.764 

45.771 

45.778 

45.785 

.6 

45.792 

45.799 

45.807 

45.814 

45.821 

45.828 

45.835 

45.842 

45.849 

_  45.856 

.7 

45.863 

45.870 

45.877 

45.884 

45.891 

45.898 

45.905 

45.912 

45.919 

45.926 

.8 

45.933 

45.940 

45.947 

45.954 

45.961 

45.968 

45.975 

45.982 

45.989 

45.996 

.9 

46.003 

46.010 

46.017 

46.024 

46.031 

46.038 

46.045 

46.052 

46.059 

46.066 

33.0 

46.073 

46.080 

46.086 

46.093 

46.100 

46.107 

46.114 

46.121 

46.128 

46.135 

.1 

46.142 

46.149 

46.156 

46.163 

46.170 

46.177 

46.184 

46.191 

46.198 

46.205 

.2 

46.212 

46.219 

46.226 

46.233 

46.240 

46.247 

46.254 

46.261 

46.268 

46.275 

.3 

46.281 

46.288 

46.295 

46.302 

46.309 

46.316 

46.323 

46.330 

46.337 

46.344 

.4 

46.351 

46.358 

46.365 

46.372 

46.379 

46.386 

46.393 

46.399 

46.406 

46.413 

.5 

46.420 

46.427 

46.434 

46.441 

46.448 

46.455 

46.462 

46.469 

46.476 

46.483 

.6 

46.489 

46.496 

46.503 

46.510 

46.517 

46.524 

46.531 

46.538 

46.545 

46.552 

.7 

46.559 

46.566 

46.572 

46.579 

46.586 

46.593 

46.600 

46.607 

46.614 

46.621 

.8 

46.628 

46.635 

46.642 

46.648 

46.655 

46.662 

46.669 

46.676 

46.683 

46.690 

.9 

46.697 

46.703 

46.710 

46.717 

46.724 

46.731 

46.739 

46.745 

46.752 

46.759 

34.0 

46.765 

46.772 

46.779 

46.786 

46.793 

46.800 

46.807 

46.814 

46.820 

46.827 

.1 

46.834 

46.841 

46.848 

46.855 

46.862 

46.868 

46.875 

46.882 

46.889 

46.896 

.2 

46.903 

46.910 

46.916 

46.923 

46.930 

46.937 

46.944 

46.951 

46.958 

46.964 

.3 

46.971 

46.978 

46.985 

46.992 

46.999 

47.005 

47.012 

47.019 

47.026 

47.033 

.4 

47.040 

47.047 

47.053 

47.060 

47.067 

47.074 

47.081 

47.088 

47.094 

47.101 

.5 

47.108 

47.115 

47.122 

47.128 

47.135 

47.142 

47.149 

47.156 

47.163 

47.169 

.6 

47.176 

47.183 

47.190 

47.197 

47.203 

47.210 

47.217 

47.224 

47.231 

47.238 

.7 

47.244 

47.251 

47.258 

47.265 

47.272 

47.278 

47.285 

47.292 

47.299 

47.306 

.8 

47.312 

47.319 

47.326 

47.333 

47.340 

47.346 

47.353 

47.360 

47.367 

47.374 

.9 

47.380 

47.387 

47.394 

47.401 

47.407 

47.414 

47.421 

47.428 

47.435 

47.441 

249 


TABLE    X  X  X  —  CONTINUED. 
VELOCITIES,  IN  FEET  PER  SECOND,  DUE  TO  HEADS  FROM  35   TO  39.99  FEET. 


Head. 

O 

1 

3 

3 

4 

5 

6 

7 

8 

9 

35.0 

47.448 

47.455 

47.462 

47.469 

47.475 

47.482 

47.489 

47.496 

47.502 

47.509 

.1 

47.516 

47.523 

47.529 

47.536 

47.543 

47.550 

47.556 

47.563 

47.570 

47.577 

.2 

47.584 

47.590 

47.597 

47.604 

47.611 

47.617 

47.624 

47.631 

47.638 

47.644 

.3 

47.651 

47.658 

47.665 

47.671 

47.678 

47.685 

47.692 

47.698 

47.705 

47.712 

.4 

47.719 

47.725 

47.732 

47.739 

47.745 

47.752 

47.759 

47.766 

47.772 

47.779 

.5 

47.786 

47.793 

47.799 

47.806 

47.813 

47.819 

47.826 

47.833 

47.840 

47.846 

.6 

47.853 

47.860 

47.867 

47.873 

47.880 

47.887 

47.893 

47.900 

47.907 

47.914 

.7 

47.920 

47.927 

47.934 

47.940 

47.947 

47.954 

47.961 

47.967 

47.974 

47.981 

.8 

47.987 

47.994 

48.001 

48.007 

48.014 

48.021 

48.028 

48.034 

48.041 

48.048 

.9 

48.054 

48.061 

48.068 

48.074 

48.081 

48.088 

48.094 

48.101 

48.108 

48.115 

36.0 

48.121 

48.128 

48.134 

48.141 

48.148 

48.155 

48.161 

48.168 

48.175 

48.181 

.1 

48.188 

48.195 

48.201 

48.208 

48.215 

48.221 

48.228 

48.235 

48.241 

48.248 

.2 

48.255 

48.261 

48.268 

48.275 

48.281 

48.288 

48.295 

48.302 

48.308 

48315 

.3 

48.321 

48.328 

48.335 

48.341 

48.348 

48T355 

48.361 

48.368 

48.375 

48.381 

.4 

48.388 

48.394 

48.401 

48.408 

48.414 

48.421 

48.428 

48.434 

48.441 

48.448 

.5 

48.454 

48.461 

48.467 

48.474 

48.481 

48.487 

48.494 

48.501 

48.507 

48.514 

.6 

48.521 

48.527 

48.534 

48.540 

48.547 

48.554 

48.560 

48.567 

48.574 

48.580 

.7 

48.587 

48.593 

48.600 

48.607 

48.613 

48.620 

48.626 

48.633 

48.640 

48.646 

.8 

48.653 

48.660 

48.666 

48.673 

48.679 

48.686 

48.693 

48.699 

48.706 

48.712 

.9 

48.719 

48.726 

48.732 

48.739 

48.745 

48.752 

48.759 

48.765 

48.771 

48.778 

37.0 

48.785 

48.792 

48.798 

48.805 

48.811 

48.818 

48.824 

48.831 

48.838 

48.844 

.1 

48.851 

48.857 

48.864 

48.871 

48.877 

48.884 

48.890 

48.897 

48.903 

48.910 

.2 

48.917 

48.923 

48.930 

48.936 

48.943 

48.950 

48.956 

48.963 

48.969 

48.976 

.3 

48.982 

48.989 

48.995 

49.002 

49.009 

49.015 

49.022 

49.028 

49.035 

49.041 

.4 

49.048 

49.055 

49.061 

49.068 

49.074 

49.081 

49.087 

49.094 

49.100 

49.107 

.5 

49.113 

49.120 

49.127 

49.133 

49.140 

49.146 

49.153 

49.159 

49.166 

49.172 

.6 

49.179 

49.185 

49.192 

49.199 

49.205 

49.212 

49.218 

49.225 

49.231 

49.238 

.7 

49.244 

49.251 

49.257 

49.264 

49.270 

49.277 

49.283 

49.290 

49.297 

49.303 

.8 

49.310 

49.316 

49.323 

49.329 

49.336 

49.342 

49.349 

49.355 

49.362 

49.368 

.9 

49.375 

49.381 

49.388 

49.394 

49.401 

49.407 

49.414 

49.420 

49.427 

49.433 

38.0 

49.440 

49.446 

49.453 

49.459 

49.466 

49.472 

49.479 

49.485 

49.492 

49.498 

.1 

49.505 

49.511 

49.518 

49.524 

49.531 

49.537 

49.544 

49.550 

49.557 

49.563 

.2 

49.570 

49.576 

49.583 

49.589 

49.596 

49.602 

49.609 

49.615 

49.622 

49.628 

.3 

49.635 

49.641 

49.648 

49.G54 

49.661 

49.667 

49.673 

49.680 

49.686 

49.693 

.4 

49.699 

49.706 

49.712 

49.719 

49.725 

49.732 

49.738 

49.745 

49.751 

49.758 

.5 

49.764 

49.770 

49.777 

49.783 

49.790 

49.796 

49.803 

49.809 

49.816 

49.822 

.6 

49.829 

49.835 

49.842 

49.848 

49.854 

49.861 

49.867 

49.874 

49.880 

49.887 

.7 

49.893 

49.900 

49.906 

49.912 

49.919 

49.925 

49.932 

49.938 

49.945 

49.951 

.8 

49.958 

49.964 

49.970 

49.977 

49.983 

49.990 

49.996 

50.003 

50.009 

50.015 

.9 

50.022 

50.028 

50.035 

50.041 

50.048 

50.054 

50.060 

50.067 

50.073 

50.080 

39.0 

50.086 

50.093 

50.099 

50.105 

50.112 

50.118 

50.125 

50.131 

50.137 

50.144 

.1 

50.150 

50.157 

50.163 

50.170 

50.176 

50.182 

50.189 

50.195 

50.202 

50.208 

.2 

50.214 

50.221 

50.227 

50.234 

50.240 

50.246 

50.253 

50.259 

50.266 

50.272 

.3 

50.278 

50.285 

50.291 

50.298 

50.304 

50.310 

50.317 

50.323 

50.330 

50.336 

.4 

50.342 

50.349 

50.355 

50.362 

50.368 

50.374 

50.381 

50.387 

50.393 

50.400 

.5 

50.406 

50.413 

50.419 

50.425 

50.432 

50.438 

50.444 

50.451 

50.457 

50.464 

.6 

50.470 

50.476 

50.483 

50.489 

50.495 

50.502 

50.508 

50.515 

50.521 

50.527 

.7 

50.534 

50.540 

50.546 

50.553 

50.559 

50.565 

50.572 

50.578 

50.585 

50.591 

.8 

50.597 

50.604 

50.610 

50.616 

50.623 

50.629 

50.635 

50.642 

50.648 

50.654 

.9 

50.661 

50.667 

50.673 

50.680 

50.686 

50.692 

50.699 

50.705 

50.712 

50.718 

32 


TABLE    XXX— CONTINUED. 
VELOCITIES,  IN  FEET  PER  SECOND,  DUE  TO  HEADS  FROM  40  TO  44.99  FEET. 


Head. 

O 

1 

a 

3 

4 

5 

6 

7 

8 

9 

40.0 

50.724 

50.731 

50.737 

50.743 

50.750 

50.756 

50.762 

50.769 

50.775 

50.781 

.1 

50.788 

50.794 

50.800 

50.807 

50.813 

50.819 

50.826 

50.832 

50.838 

50.845 

.2 

50.851 

50.857 

50.863 

50.870 

50.876 

50.882 

50.889 

50.895 

50.901 

50.908 

.3 

50.914 

50.920 

50.927 

50.933 

50.939 

50.946 

50.952 

50.958 

50.965 

50.971 

.4 

50.977 

50.983 

50.990 

50.996 

51.002 

51.009 

51.015 

51.021 

51.028 

51.034 

.5 

51.040 

51.047 

51.053 

51.059 

51.065 

51.072 

51.078 

51.084 

51.091 

51.097 

.6 

51.103 

51.110 

51.116 

51.122 

51.128 

51.135 

51.141 

51.147 

51.154 

51.160 

.7 

51.166 

51.172 

51.179 

51.185 

51.191 

51.198 

51.204 

51.210 

51.216 

51.223 

.8 

51.229 

51.235 

51.241 

51.248 

51.254 

51.260 

51.267 

51.273 

51.279 

51.285 

.9 

51.292 

51.298 

51.304 

51.310 

51.317 

51.323 

51.329 

51.336 

51.342 

51.348 

41.0 

51.354 

51.361 

51.367 

51.373 

51.379 

51.386 

51.392 

51.398 

51.404 

51.411 

.1 

51.417 

51.423 

51.429 

51.436 

51.442 

51.448 

51.454 

51.461 

51.467 

51.473 

.2 

51.479 

51.486 

51.492 

51.498 

51.504 

51.51t 

51.517 

51.523 

51.529 

51.536 

.3 

51.542 

51.548 

51.554 

51.561 

51.567 

51.573 

51.579 

51.586 

51.592 

51.598 

.4 

51.604 

51.610 

51.617 

51.623 

51.629 

51.635 

51.642 

51.648 

51.654 

51.660 

.5 

51.667 

51.673 

51.679 

51.685 

51.691 

51.698 

51.704 

51.710 

51.716 

51.723 

.6 

51.729 

51.735 

51.741 

51.747 

51.754 

51.760 

51.766 

51.772 

51.778 

51.785 

.7 

51.791 

51.797 

51.803 

51.809 

51.816 

51.822 

51.828 

51.834 

51.841 

51.847 

.8 

51.853 

51.859 

51.865 

51.872 

51.878 

51.884 

51.890 

51.896 

51.903 

51.909 

.9 

51.915 

51.921 

51.927 

51.934 

51.940 

51.946 

51.952 

51.958 

51.964 

51.971 

42.0 

51.977 

51.983 

51.989 

51.995 

52.002 

52.008 

52.014 

52.020 

52.026 

52.032 

.1 

52.039 

52.045 

52.051 

52.057 

52.063 

52.070 

52.076 

52.082 

52.088 

52.094 

.2 

52.100 

52.107 

52.113 

52.119 

52.125 

52.131 

52.137 

52.144 

52.150 

52.156 

.3 

52.162 

52.168 

52.174 

52.181 

52.187 

52.193 

52.199 

52.205 

52.211 

52.218 

.4 

52.224 

52.230 

52.236 

52.242 

52.248 

52.255 

52.261 

52.267 

52.273 

52.279 

.5 

52.285 

52.291 

52.298 

52.304 

52.310 

52.316 

52.322 

52.328 

52.334 

52.341 

.6 

52.347 

52.353 

52.359 

52.365 

52.371 

52.377 

52.384 

52.390 

52.396 

52.402 

.7 

52.408 

52.414 

52.420 

52.427 

52.433 

52.439 

52.445 

52.451 

52.457 

52.463 

.8 

52.470 

52.476 

52.482 

52.488 

52.494 

52.500 

52.506 

52.512 

52.519 

52.525 

.9 

52.531 

52.537 

52.543 

52.549 

52.555 

52.561 

52.567 

52.574 

52.580 

52.586 

43.0 

52.592 

52.598 

52.604 

52.610 

52.616 

52.623 

52.629 

52.635 

52.641 

52.647 

.1 

52.653 

52.659 

52.665 

52.671 

52.678 

52.684 

52.690 

52.696 

52.702 

52.708 

.2 

52.714 

52.720 

52.726 

52.732 

52.738 

52.745 

52.751 

52.757 

52.763 

52.769 

.3 

52.775 

52.781 

52.787 

52.793 

52.799 

52.806 

52.812 

52.818 

52.824 

52.830 

.4 

52.836 

52.842 

52.848 

52.854 

52.860 

52.866 

52.873 

52.879 

52.885 

52.891 

.5 

52.897 

52.903 

52.909 

52.915 

52.921 

52.927 

52.933 

52.939 

52.945 

52.952 

.6 

52.958 

52.964 

52.970 

52.976 

52.982 

52.988 

52.994 

53.000 

53.006 

53.012 

.7 

53.018 

53.024 

53.030 

53.037 

53.043 

53.049 

53.055 

53.061 

53.067 

53.073 

.8 

53.079 

53.085 

53.091 

53.097 

53.103 

53.109 

53.115 

53.121 

53.127 

53.133 

.9 

53.139 

53.146 

53.152 

53.158 

53.164 

53.170 

53.176 

63.182 

53.188 

53.194 

44.0 

53.200 

53.206 

53.212 

53.218 

53.224 

53.230 

53.236 

53.242 

53.248 

53.254 

.1 

53.260 

53.266 

53.272 

53.279 

53.285 

53.291 

53.297 

53.303 

53.309 

53.315 

.2 

53.321 

53.327 

53.333 

53.339 

53.345 

53.351 

53.357 

53.363 

53.369 

53.375 

.3 

53.381 

53.387 

53.393 

53.399 

53.405 

53.411 

53.417 

53.423 

53.429 

53.435 

.4 

53.441 

53.447 

53.453 

53.459 

53.465 

53.471 

53.477 

53.483 

53.489 

53.495 

.5 

53.501 

53.507 

53.513 

53.519 

53.525 

53.531 

53.537 

53.543 

53.549 

53.555 

.6 

53.561 

53.567 

53.573 

53.579 

53.586 

53.592 

53.598 

53.604 

53.610 

53.616 

.7 

53.621 

53.627 

53.633 

53.639 

53.645 

53.651 

53.657 

53.663 

53.669 

53.675 

.8 

53.681 

53.687 

53.693 

53.699 

53.705 

53.711 

53.717 

53.723 

53.729 

53.735 

.9 

53.741 

53.747' 

53.753 

53.759 

53.765 

53.771 

53.777 

53.783 

53.789 

53.795 

251 


TABLE    X  X  X  —  CONTINUED. 
VELOCITIES,  IN  FEET  PER  SECOND,  DUE  TO  HEADS  FROM  45  TO  49.99  FEET. 


Head. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 

45.0 

53.801 

53.807 

53.813 

53.819 

53.825 

53.831 

53.837 

53.843 

53.849 

53.855 

.1 

53.861 

53.867 

53.873 

53.879 

53.885 

53.891 

53.897 

53.903 

53.909 

53.915 

.2 

53.921 

53.927 

53.932 

53.938 

53.944 

53.950 

53.956 

53.962 

53.968 

53.974 

.3 

53.980 

53.986 

53.992 

53.998 

54.004 

54.010 

54.016 

54.022 

54.028 

54.034 

.4 

54.040 

54.046 

54.052 

54.058 

54.064 

54.069 

54.075 

54.081 

54.087 

54.093 

.5 

54.099 

54.105 

54.111 

54.117 

54.123 

54.129 

54.135 

54.141 

54.147 

54.153 

.6 

54.159 

54.165 

54.170 

54.176 

54.182 

54.188 

54.194 

54.200 

54.206 

54.212 

.7 

54.218 

54.224 

54.230 

54.236 

54.242 

54.248 

54.254 

54.259 

54.265 

54.271 

.8 

54.277 

54.283 

54.289 

54.295 

54.301 

54.307 

54.313 

54.319 

54.325 

54.331 

.9 

54.336 

54.342 

54.348 

54.354 

54.360 

54.366 

54.372 

54.378 

54.384 

54.390 

46.0 

54.396 

54.402 

54.407 

54.413 

54.419 

54.425 

54.431 

54.437 

54.443 

54.449 

.1 

54.455 

54.461 

54.467 

54.472 

54.478 

54.484 

54.490 

54.496 

54.502 

54.508 

.2 

54.514 

54.520 

54.526 

54.531 

54.537 

54.543 

54.549 

54.555 

54.561 

54.567 

.3 

54.573 

54.579 

54.585 

54.590 

54.596 

54.602 

54.608 

54.614 

54.620 

54.626 

.4 

54.632 

54.638 

54.643 

54.649 

54.655 

54.661 

54.667 

54.673 

54.679 

54.685 

.5 

54.690 

54.696 

54.702 

54.708 

54.714 

54.720 

54.726 

54.732 

54.737 

54.743 

.6 

54.749 

54.755 

54.761 

54.767 

54.773 

54.779 

54.784 

54.790 

54.796 

54.802 

.7 

54.808 

54.814 

54.820 

54.826 

54.831 

54.837 

54.843 

54.849 

54.855 

54.861 

.8 

54.807 

54.872 

54.878 

54.884 

54.890 

54.896 

54.902 

54.908 

54.913 

54.919 

.9 

54.925 

54.931 

54.937 

54.943 

54.949 

54.954 

54.960 

54.966 

54.972 

54.978 

47.0 

54.984 

54.990 

54.995. 

55.001 

55.007 

55.013 

55.019 

55.025 

55.030 

55.036 

.1 

55.042 

55.048 

55.054 

55.060 

55.066 

55.071 

55.077 

55.083 

55.089 

55.095 

.2 

55.101 

55.106 

55.112 

55.118 

55.124 

55.130 

55.136 

55.141 

55.147 

55.153 

.3 

55.159 

55.165 

55.171 

55.176 

55.182 

55.188 

55.194 

55.200 

55.206 

55.211 

.4 

55.217 

55.223 

55.229 

55.235 

55.240 

55.246 

55.252 

55.258 

55.264 

55.270 

.5 

55.275 

55.281 

55.287 

55.293 

55.299 

55.304 

55.310 

55.316 

55.322 

55.328 

.6 

55.334 

55.339 

55.345 

55.351 

55.357 

55.363 

55.368 

55.374 

55.380 

55.386 

.7 

55.392 

55.397 

55.403 

55.409 

55.415 

55.421 

55.426 

55.432 

55.438 

55.444 

.8 

55.450 

55.455 

55.461 

55.467 

55.473 

55.479 

55.484 

55.490 

55.496 

55.502 

.9 

55.508 

55.513 

55.519 

55.525 

55.531 

55.537 

55.542 

55.548 

55.554 

55.560 

48.0 

55.566 

55.571 

55.577 

55.583 

55.589 

55.595 

55.600 

55.606 

55.612 

55.618 

.1 

55.623 

55.629 

55.635 

55.641 

55.647 

55.652 

55.658 

55.664 

55.670 

55.675 

.2 

55.681 

55.687 

55.693 

55.699 

55.704 

55.710 

55.716 

55.722 

55.727 

55.733 

.3 

55.739 

55.745 

55.750 

55.756 

55.762 

55.768 

55.774 

55.779 

55.785 

55.791 

.4 

55.797 

55.802 

55.808 

55.814 

55.820 

55.825 

55.831 

55.837 

55.843 

55.848 

.5 

55.854 

55.860 

55.866 

55.872 

55.877 

55.883 

55.889 

55.895 

55.900 

55.906 

.6 

55.912 

55.918 

55.923 

55.929 

55.935 

55.941 

55.946 

55.952 

55.958 

55.964 

.7 

55.969 

55.975 

55.981 

55.987 

55.992 

55.998 

56.004 

56.009 

56.015 

56.021 

.8 

56.027 

56.032 

56.038 

56.044 

56.050 

56.055 

56.061 

56.067 

56.073 

56.078 

.9 

56.084 

56.090 

56.096 

56.101 

56.107 

56.113 

56.118 

56.124 

56.130 

56.136 

49.0 

56.141 

56.147 

56.153 

56.159 

56.164 

56.170 

56.176 

56.181 

56.187 

56.193 

.1 

56.199 

56.204 

56.210 

56.216 

56.222 

56.227 

56.233 

56.239 

56.244 

56.250 

.2 

56.256 

56.262 

56.267 

56.273 

56.279 

56.284 

56.290 

56.296 

56.302 

56.307 

.3 

56.313 

56.319 

56.324 

56.330 

56.336 

56.342 

56.347 

56.353 

56.359 

56.364 

.4 

56.370 

56.376 

56.381 

56.387 

56.393 

56.399 

56.404 

56.410 

56.416 

58.421 

.5 

56.427 

56.433 

56.439 

56.444 

56.450 

56.456 

56.461 

56.467 

56.473 

56.478 

.6 

56.484 

56.490 

56.495 

56.501 

56.507 

56.513 

56.518 

56.524 

56.530 

56.535 

.7 

56.541 

56.547 

56.552 

56.558 

56.564 

56.569 

56.575 

56.581 

56.586 

56.592 

.8 

56.598 

56.604 

56.609 

56.615 

56.621 

56.626 

56.632 

56.638 

56.643 

56.649 

.9 

56.655 

56.660 

56.666 

56.672 

56.677 

56.683 

56.689 

56.694 

56.700 

56.706 

10    f'rrt 


flaatet  /tar/frfy  M ' '. 


; . 


fy  TJltfoHand  A  Co 


IKMIUiM 


i  i\ n I T\L . 


Pan  ffarf/Mi, 


lid.  9 


fio    u 


Fio.  5. 


Scale  tor   Figure's  1.2.5  and  4 


J  6' 


J 1 1 '+• 


Scfde  for   Figures  5.  (>,  7  and  8  . 


Scale  for  FiQure*  9.1°  and  11. 


Fio .  (') 


APPARATUS    USED    IN    GAUGING    THI 


Fio 


\ 


Seftle    to'1    Fiourcs 

.1        ~    j 

— I 1 1 — 


1.  '.!.  .").  .'),  (\  7.  rt.  ;)  a  nil   JO 
,i  t>~  s 


tor   Fio    4 


Fui  (i  Fiii  .'> 


Fi.i 


Fio  4- 


• 


YATER  AT    THE   TREMONT    TTRB1NK 


- 


Am    //a<"//r'-/i/ 


u.  i  mini  .u  u      i  k'uu      i  LI       i  ui,       i 


"UN.     OT     A     iT 


rOTHE   VENT    WHEEL  AT 


•,    1WOTT    dlTTOX     Mfl.l.S. 


\ 


I-J... } . 
±trmj: 
fffl  t 


i-ir 


-r  4-f-i 

rtrf  h 


Ukf-L 


EfS 

-H-  1-H- 


H  -H-H-H- 


Fi«  I. 


Kid   .Y 


Fid.  (i. 


Fid    H: 


Fid    !."> 


Fio.i4. 


Fki   5. 


Fio   4.. 


Fid.  «. 


Tid.u. 


Kg.  12. 


L,J\I  ij  u 1  m  LJ  a  i  o       rv  i 


LOVVKK.    LIHKS 


Pi.  \l. 


EXPERIMENTS    UPON    WE 


''a       a       a       a       a       a  , 


>    AT    THF,    f,0\VF,R    LOCKS 


'• 


KKI'F.IUMLNTS    I'l'llN    WF.I 


Scale   for  Figures  '3,3  and  4' 


So.ilo     tor    Fio  .  1 . 


Pan 


AT   TIIK    I.OWKIl    LOCKS 


.xm. 


VN  i  r. \>     ui 


Inr,     I, UVVr.lv     MM  l\> 


m.  4.. 


I.Q         I    I  j'   '  > 

I) 

A 


r 


Hiram  ?.  .1,11,    ,M 


B    E 


Fid.  H. 


Kid.  <i. 


Fid, 


Sralc      li.l-     Flos     .").  II.  -.    II,  ;,    and    10 





i    and    3 . 


'''/    i^lnll';; 
-iff® 


yiijiiuiuLLUJJi 


1 


ii 


E 

,  — 

^  -"   ^  -if-'^^^^r-  ^-;::,^4 

.  . 


Mram  /-'.JIM.r  M 


WATKR   IN  OPKN    CANALS 


PI.  XVI 


r] 

r 

/  1 

r 

\ 

i) 

i) 

,;  i 

1) 

D 

vi 

j 

j 

—  4-- 

\ 

i 

I) 

id 

' 

/ 

Kxp.  i. 


Exp.  laS.' 


o 


Exp   68. 


20  1.3.  3f2          t> 


Exp.  7 


Ixp .  12$ ._ 


EXP.  6 


Exp.g-J. 


Tlie     ABSC1 

The.     ORD1I 
The      frt/tn 

of    Hie     tu 


j.l. hn      a 


Exp.  ;)(>. 


S     .of    the     /mints     rn,ti  //,;/     o     represent      III,'     me,  in      ,//.r/,if/e,:r     in  Ji-et     ,if    Me      ,ret>er,ii     ti/be.r.  Jh»u     t/ic      /e/i 


'.S       ,-/'     til,'      [>,iin/.r      tn,i,  •/,;;/  n-finvt'til     ///,•      t>et«,-Hi<v     ,>/'    //!<•      .rein-rill.     Ilil',:-      in   _  /',;>!     per      set-tin,!  . 

i>/'    t/ie      irregularly     firrt>r</  /ine.r.    i;-fir<\ffnl     the     ni<;i/i      ne/<'i'i/i<:r     n/'     Hie      ////',  :i      in   Ji;-l   [>er     .reeiitiil     in 

f.n-t.r     ,>/'    th<'     n>i<//h      i//'    /hi'  J/unie  . 

•<i/e    ii  f     ,t//    the     ,/i<i,/ni//i.r     i'//  t/n.r    /j/,i/,-  ,    i.r     ,1/1,-     /iiiir/Ji     ,if    //nil     ;'/'  the     i>ru/inat.r    /!wn     n>/neh    the    nieiiii    ne/tieiiie.f 

have       bi'fii     determined  . 


10  G. 


Lxp.g/. 


/t>       /.i  .172 


Exp.   ing. 


Kxp.    u< 


3.0 


3,5 


Exp. 


*  ' 


22  23  21  2S 


27  2S  29  30  HI  32  S3  3k  3S  36  31  3ft  39  Ifff 


EXPERIMENTS    ON    THE    FLOW   OF    WATER   TflROU 


SUBMERGED    ORIFICES    AND    DIVERGING    TUBES. 


PL  XX. 


EXPERIMENTS  ON  THF.   FLOW   OF   WATER .  TIIKOn 


Fio.ii.       F 


Scale    except    for    Figures    8    and 


SPBMKKOKI)   OR1FICKS    AND   DIVKKGINC.   Tl  II KS 


TURBINE   WATER  -  WHEK: 


>F    700    HORSE   POWF.R 


PI.  xxn. 


Srale    for     liourc    I. 


Srah-    la  I'    fidllivs      •_>    ft  7)  . 


TURBM  WATER- WEE1 


700  HORSF,    I'OWKK 


\ 


UNIVKRKITY  OP  CALIFORNIA  LIBRARY 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


OCT  13  , 


y?ODISCMARu6'90 


957 


30m-l,'15 


U.C.  BERKELEY  LIBRARIES 


fflBRfflJHHBHHnBB 

$${$$m(l$w 


Illilllli 


